Handbooks in Central Banking

No. 23


Emilio Fernandez-Corugedo

Series editors: Andrew Blake & Gill Hammond Issued by the Centre for Central Banking Studies, Bank of England, London EC2R 8AH E-mail: [email protected] July 2004 © Bank of England 2004 ISBN 1 85730 143 9

Consumption Theory Emilio Fernandez-Corugedo1 Centre for Central Banking Studies, Bank of England


This Handbook represents the views and analysis of the author and should not be thought to represent those of the Bank of England or Monetary Policy Committee members. The Handbook is a modified chapter from my Bristol University PhD thesis ‘Essays on Consumption’. I would like to thank my PhD advisers Cliff Attfield, Nigel Duck and David Demery for their insights. Suggestions from Andrew Benito, Andrew Blake, Gill Hammond and Paul Robinson greatly improved the Handbook. All errors and omissions remain, of course, my responsibility.

Contents 1 Introduction


2 Consumption: key theories and terminology 2.1 The absolute income hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The permanent income hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The life cycle hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3 5

Box A: Understanding consumption terminology


3 Rational expectations, unit roots and poor forecasting


4 The rational expectations permanent income hypothesis


Box B: The impact of interest rates on consumption


Box C: The role of assumptions in REPI


5 Departing from REPI 5.1 Breaking certainty equivalence: precautionary saving . . . . . . . . 5.1.1 The buffer stock model . . . . . . . . . . . . . . . . . . . . 5.2 Near rationality and aggregation . . . . . . . . . . . . . . . . . . . 5.2.1 Near rationality and the updating of consumption decisions 5.2.2 Near rationality and imperfect information . . . . . . . . . 5.3 Liquidity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Habits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Finite lives and REPI . . . . . . . . . . . . . . . . . . . . . . . . .

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Box D: Risk aversion and prudence

13 14 15 17 17 19 20 22 23 25

6 Other consumption issues 27 6.1 Durables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Aggregation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 Conclusions and suggestions for further reading


A The derivation of the Euler equation


B Innovations in consumption and labour income: smoothness B.1 Testing for excess sensitivity and smoothness . . B.1.1 Excess sensitivity . . . . . . . . . . . . . . B.1.2 Excess smoothness . . . . . . . . . . . . .

tests for excess sensitivity and 33 . . . . . . . . . . . . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . 34

C The substitution, income and wealth revaluation effects; the two period case with diagrams 35 D Taylor-series expansion of the habit model




Introduction ‘Consumer expenditure accounts for between 50% and 70% of spending in most economies. Not surprisingly, the consumption function has been the most studied of the aggregate expenditure relationships and has been a key element of all the macroeconomic model building efforts since the seminal work of Klein and Goldberger (1955).’ Muellbauer and Lattimore (1994, p. 292).

There is no doubt that aggregate consumption is a key variable for policy makers. The aim of this handbook is to familiarise the reader with the key theories that have been used to model and forecast consumption and draw out their implications for policy analysis.1 This handbook is intended to be accessible to those working in policy-related departments without losing economic rigour. Important concepts are highlighted in a series of four boxes and technical details in four appendices. The handbook pays particular attention to the role of forward-looking agents and their reaction to policy announcements; the role of interest rates in consumption and the role of other variables thought to affect consumption behaviour such as taxes, the structure of both the banking system and the stock market, age and wealth distributions and the volatility of economic variables. Unfortunately, different consumption theories can lead to different policy prescriptions and as such a clear message arises from this handbook: there is no single consumption theory that can explain consumption behaviour in all economies; economists must therefore investigate what they think explains consumption in their country.


Consumption: key theories and terminology

Familiarity with modern consumption research requires understanding three fundamental models: Keynes’s (1936) absolute income hypothesis (AIH), Friedman’s (1957) permanent income hypothesis (PIH), and Modigliani’s (1949) life-cycle hypothesis (LCH). Modern consumption research is based to varying degrees on at least one of these approaches.


The absolute income hypothesis

Macroeconomic research on the aggregate consumption function is thought to have began with the publication of Keynes’s (1936)2 principle that consumption (expenditure) was a stable, not necessarily linear, function of disposable income: ct = α + βyt


where ct and yt denote the real values of total personal consumption expenditure and total disposable income respectively at time t. The marginal propensity to consume (mpc), β, was expected to be constant and close to one, and α, the autonomous component of consumption, was assumed to be small but positive. Keynes argued that the average propensity to consume (apc), yc , would exceed the mpc, so that the income elasticity of consumption defined as mpc apc , would be less than unity 1

Some issues related to the consumption literature will not be developed fully (they may be relegated to footnotes); for instance the consumption and asset pricing models (CAPM), the joint labour supply and consumption choice within general equilibrium models (including those which also examine the demand for money). Throughout, we shall assume a partial equilibrium world where consumers make decisions subject to the evolution of their financial wealth and labour income. 2 Although we should not disregard excellent earlier work by Ramsey (1928) and Fischer (1930).


(although it would approach unity as income increased). Hence in the long-run, in the face of income growth, one would expect the income elasticity to be unity.3 The implications of (1) for policy makers are extremely simple; an increase in aggregate (after tax) income yields an increase in consumption. Nothing else affects consumption expenditure: there is not a specific role for the interest rate, money nor the exchange rate. Thus, changes in the instruments we normally think a central bank controls cannot, on their own, affect consumption expenditure directly. Of course, if the instruments available to central banks are able to affect disposable income, then according to the AIH, they will also affect consumption indirectly (and probably with a lag thereby introducing lags of monetary policy on the economy4 ). Empirical research found that (1) could not explain aggregate data.5, 6 In view of the evidence, economists attempted to correct the inadequacy of (1). Two theories had a lasting effect on consumption research: Modigliani’s (1954) life cycle hypothesis (LCH), and Friedman’s (1957) permanent income hypothesis (PIH).


The permanent income hypothesis

The PIH represents an important development of Keynes’s aggregate consumption function. Unlike (1), the PIH was inspired by microfoundations and representative agents, and highlighted the importance of not just the present but also the future.7 The fundamental building-block of PIH is that individuals want to maximise their lifetime well-being (utility) subject to the constraint that all their lifetime resources must be spent. The PIH focused on distinguishing between consumption and current expenditure on the one hand, and income and current receipts on the other. Because consumers are thought to plan their expenditures not on the basis of income received during the current period but rather on the basis of income expected during their lifetime, consumers plan their expenditure on the grounds of a long-run view of the resources that will be available to them. Friedman postulated that income, y, is made up of two components: a permanent component p (y ) and a transitory component (y t ).8 Friedman argued that some of the factors that give rise to the transitory component of income were specific to particular consumers (illnesses, bad harvests, etc.) but that for any considerable group of consumers the transitory components tend to average 3 The actual ‘linear’ form of (1) is not without qualifications and should not be interpreted literally. Keynes acknowledged that unexpected changes in capital values, substantial changes in the rate of interest as well as changes in the distribution of income could have significant influences upon the mpc. Keynes also added that as a rule, the proportion of income saved tends to increase with income but he did not consider that remark a fundamental psychological law. Finally, Keynes recognised that because of habit persistence and slow-adjustment the long-run mpc was likely to be larger than the short-run propensity. 4 The impact of changes in monetary policy on macroeconomic variables is known as the transmission mechanism. 5 This is not to say that Keynes’s early work did not capture important aspects of consumption behaviour: the income term appeared to account for most of the variability in consumption (with a coefficient less than one) and the apc was larger than the mpc. 6 Reasons given for the failure of (1) were: i) the presence of a deterministic trend in α could not be ruled out, ii) the apc did not contain a significant trend, iii) α had a tendency to shift upwards in time, and iv) estimates of β were lower than predicted by the theory. A number of post-war studies also pointed out that the absolute income hypothesis could not explain the commonly observed fact that the apc had remained constant in the US since the 1870s when cross-section data at various points in time indicated that the mpc declined as incomes rose. 7 See Box A for definitions of important concepts such as utility, indifference curves and budget constraints. 8 The permanent component was interpreted ‘as reflecting the effect of those factors that the unit regards as determining its capital value or wealth; the non-human wealth it owns; the personal attributes of the economic activity of the earners in the unit, such as their training, ability, personality; the attributes of the economic activity of the earners, such as the occupation followed, the location of the economic activity, and so on.’ [p. 21]. The transitory component was interpreted ‘as reflecting all “other” factors, factors that are likely to be treated by the unit affected as “accidental” or “chance” occurrences, though they may, from another point of view, be the predictable effect of specificable forces, for example, cyclical fluctuations in economic activity.’ [pp. 21-2]


out, so that the mean of the transitory component is expected to be zero.9 Consumption expenditures also comprise permanent (cp ) and transitory components (ct ). The permanent component relates to the amount that consumers plan to consume to maximise their lifetime utility. Without uncertainty, total consumption would be equal to cp . ct relates to all ‘other’ factors. In its ‘most general form’ PIH is given by: cp = k (r, w, u) × y p p

y = y +y



c = cp + ct where letters without a superscript denote current values, r is the rate of interest at which the consumer can borrow or lend, w is the ratio of wealth to income and u refers to consumers’ taste preferences. The first equation in (2) defines the relationship between permanent consumption and permanent income. The ratio between the two variables – the mpc out of permanent income, k(·) – is independent of the size of permanent income but it does depend on other variables: r, w and u. Thus, permanent consumption has a constant marginal propensity to consume with respect to permanent income, but at the same time, that propensity to consume is allowed to deviate when any of the ceteris paribus assumptions are breached. The last two equations provide a means of linking actual measured variables (c, y) to their ‘relevant’ permanent components. To close the model, permanent income is defined. Friedman assumed adaptive expectations ¡ ¢ ytp = rρ yt + ρyt−1 + ρ2 yt−2 + · · ·

where ρ = (1 + r)−1 . Thus, permanent income is approximated by a geometrically declining weighted average of current and past actual incomes. A number of implications for policy makers arise from the PIH. Unlike (1), expectations about future variables play a crucial role; consumers do not care about the past, they only care about the present and the future.10 Moreover, if yt = ct = 0 (which will not always be the case), then (2) states that consumption is a constant proportion of income. However, note that this constant ratio is a function of the interest rate, allowing a role for central banks to affect consumption through changes in interest rates.11 Moreover, the source of the shocks can be important: consumers will react differently if a shock is permanent rather than transitory. Consumption ought to respond more forcefully to permanent rather than transitory shocks. Thus, for policy makers it is very important to be able to discern the persistence of shocks; an announcement that a reduction in taxes or interest rates will be permanent would lead to different behaviour compared to an announcement where tax or interest rate changes are thought to be only transitory. Finally, note that data definitions now become very important; consumption functions should not be formulated in terms of consumption expenditures and disposable income, but in terms of permanent and transitory consumption and income. This necessitates the use of proxies for the permanent and transitory components of income and consumption. Unfortunately, this has proved elusive for economists and provides a strong obstacle for the practical use of these kind of models. 9

Friedman himself argues that the mean of the transitory component need not be equal to zero, however he states at the same time that ‘the mean measured of the group would equal the mean permanent component’ [p. 22]. 10 However, with adaptive expectations, expectations about the future are approximated with the past and therefore past values matter. Please see section 4 to see how rational expectations make consumers more forward looking. 11 Note however, that if we also take the definition of permanent income suggested by Friedman, then interest rates will not only affect consumption through the marginal propensity to consume out of permanent income, but they will affect permanent income too. Formally, this last effect on permanent income is a wealth effect (see Box B below for more on the impact of interest rates on consumption).



The life cycle hypothesis

Modigliani’s LCH also considered forward-looking consumers who maximise lifetime utility but emphasised the life cycle evolution of both income and household consumption.12 The major difference between PIH and LCH is that the latter recognised ‘the finite life of households, (so that the LCH) could focus on those systematic variations in income and in “needs” which occur over the life cycle, as a result of maturing and retiring, and of changes in family size. In addition the LCH was in a position to take into account bequests and the bequest motive’ [Modigliani (1986, p. 300)]. In the LCH consumers maximise utility subject to the lifetime resources available to them. The consumption plan that results is a function of resources available, the rate of return on capital and the age of the agent. After making a number of assumptions, Modigliani arrived at13 ct = α1 yt + α2 yte + α3 At−1


where c represents aggregate consumption, y represents current non-property income, y e is ‘expected annual non-property income’, and A represents net worth. The model yielded a number of implications summarised in Modigliani’s (1986) Nobel Prize Lecture: 1. The saving rate of a country is entirely independent of its per capita income. 2. Differing national saving rates are consistent with an identical individual life cycle behaviour. 3. Between countries with identical individual behaviour, the aggregate saving rate will be higher the higher the long-run growth rate of the economy. It will be zero for zero growth. 4. The wealth-income ratio is a decreasing function of the growth rate, thus being largest at zero growth. 5. An economy can accumulate a very substantial stock of wealth relative to income even if no wealth is passed on by bequests. 6. The main parameter that controls the wealth-income ratio and the saving rate for given growth is the prevailing length of retirement. For policy makers, many of the implications discussed in the previous section apply to the lifecycle hypothesis (eg, forward-looking agents, the role of interest rates on consumption decisions). There are however differences; the age distribution of the economy, the age of (compulsory) retirement, life-expectancy, etc are likely to be important variables explaining consumption behaviour. These issues are discussed in more detail below, see section 5.5. 12

For the UK and the US, we normally observe hump-shaped profiles for lifetime consumption and income; that is income and consumption profiles vary throughout the lifetime of every consumer. 13 The assumptions made were: i) a homogeneous utility function, ii) no bequest motive, iii) perfect capital markets, iv) all households in the economy have the same utility functions and use the same discount rate, v) the age distribution, the age distribution of income, and the age distribution of net worth are constant, vi) expected income is proportional to current income, vii) the allocation of consumption is not affected by changes in the degree of uncertainty regarding expectations about future earnings, viii) the planning horizon of the individual household is the whole of the life-span, ix) the rate of time-preference is constant and x) the actions of the individual conform to his lifetime plans for consumption.


Box A: Understanding consumption terminology This box explains a number of terms that are used in this handbook and that are commonly found in the consumption literature.a • Utility – Individuals are able to rank situations and goods from the most desirable to the least. This ranking was termed utility by Jeremy Bentham in the book ‘Introduction to the Principles of Morals and Legislation’ (1848). Following Bentham, economists say that more desirable situations yield more utility than do less desirable ones. For intertemporal consumption problems utility is represented by the following function, termed a utility function (4) U = V (c0 , c1 , ..., cT ) Where U is termed lifetime utility and V (.) is a function with arguments c0 , c1 , ..., cT . Consumers obtain satisfaction from consuming quantities c0 , c1 , ..., cT , where c0 refers to consumption at time 0, c1 to consumption at time 1 and so on. It is assumed that agents cease to consume at T + 1. • Marginal utility – Marginal utility refers to the extra amount of utility that is derived from the consumption of an extra unit of a good or a bundle of goods. In terms of (4) the marginal utility of consuming more of c0 is given by ∂V (c0 , c1 , ..., cT ) ∂U = ∂c0 ∂c0


that is, it is equal to the partial derivative of V (.) with respect to c0 . • Strongly intertemporally additive preferences – If we assume strongly intertemporally separable utility, (4) takes the following form 1 u (c1 ) + · · · + U = u (c0 ) + 1+δ


1 1+δ


u (cT )


Lifetime utility is a function of individual u(.), where each u(.) is a function of consumption in one period only. The functions u(.) are sometimes termed ‘felicity’, ‘subutility’ or period utility functions to distinguish them from U , the lifetime utility function. δ is the rate of time preference and acts like an interest rate; it discounts future utility so that the lifetime utility function represents utility in ‘present value’ terms. • Indifference curve – An indifference curve shows a set of consumption bundles among which the consumer has the same level of utility and is therefore indifferent. Along an indifference curve, bundles of goods provide the same level of utility. The slope of an indifference curve tells us how consumers trade bundles of goods. Indifference curves are normally assumed to be downward sloping. The negative slope shows that consumers who must give up a unit of good X must be compensated by more units of a different good, say Y, to remain indifferent between the two bundles of goods. a

Many of the concepts in this box follow from intermediate microeconomic textbooks such as Nicholson (1992), Varian (1992), Kreps (1990).


• Marginal rate of substitution – The marginal rate of substitution (MRS) is the negative of the slope of an indifference curve. It tells us the answer to the following question: how many units of a good X can a consumer give up in order to consume more units of a different good say Y without changing the overall level of utility? Mathematically M RS = −

dY |Z=Z1 dX


where the notation states that the slope is to be calculated along the Z1 indifference curve such that utility is the same. • Elasticity of substitutiona – The elasticity of substitution measures how ‘difficult’ it is to substitute consumption between two periods. Such ‘difficulty’ is a question about the shape of the indifference curve. We say that substitution between periods is easy if the slope of the indifference curve does not change much as we change the ratio of consumption between two periods. Likewise, if the slope of the indifference curve does change as the ratio of consumption between periods changes, we say that substitution is difficult. Economists normally assume that indifference curves have a negative slope and are strictly convex (this guarantees balanced bundles). Such assumption implies that the slope of the indifference curve (the MRS) decreases as we move along an indifference curve by reducing consumption at 1 and increasing it at 0 (see Appendix C). If the MRS does not change for changes in cc10 , we may say that substitution is easy as the consumption mix does not affect the ratio of marginal utilities much. Alternatively, if the the MRS does change for changes in cc10 , we may say that substitution is difficult as small changes in the the consumption mix will have large effects on the ratio of marginal utilities. The mathematical formulae for the elasticity of substitution is given byb ³ ´ d cc10 (M RS) · ³ ´ θ = c1 d (M RS) c0 µ 0 ¶ u (c0 ) = u00 (c0 ) c0 The elasticity of substitution varies between θ = ∞ and θ = 0. θ = ∞ occurs when the utility 1 )b · c1 for any a and b, since u00 (ct ) = 0. Note that with function is linear, ie U(c0 , c1 ) = a · c0 + ( 1+δ linear utility functions, the indifference curve will also be linear and therefore will have a constant MRS; in other words, the ratio of marginal utilities is constant along the indifference curve as the ratio cc10 varies. Thus substitution is very easy and agents will substitute between consumption in different periods whenever they receive incentives to do so, namely changes to the i rate of interest. h 1 θ = 0 is given by a utility function of the type U (c0 , c1 ) = min a · c0 , ( 1+δ )b · c1 . In this case, the indifference curve is L−shaped, and substitution of consumption between periods is impossible.c In the literature, we normally find that the elasticity of substitution for consumption is less than 1 but greater than zero. a

For an introduction to the elasticity of substitution between inputs in a production function, see Nicholson (1992, pp. 307-308). b The last equality is true if we let time 1 approach time 0. c Common examples where an L-shaped function might be used are: a utility function for left and right hand goods (gloves, shoes, etc.) or a production function such as a machine that can only be operated by one worker. In both cases, increasing the quantity of one of the goods/inputs (say one more left glove, or one more worker) will not increase utility or production unless more of the other goods/inputs is increased at the same time (one more right glove or one more machine).


• The budget constraint – A consumer’s budget constraint shows how a consumer’s lifetime resources can be allocated for the purchase of lifetime consumption bundles. The ‘period budget constraint’ is normally given by A1 = (1 + r0 ) (A0 + w0 − c0 )


where A denote assets, w is labour income, r is the rate of interest and c is consumption. This expression states that the level of assets in period 1, A1 , is equal to the difference between all the resources available for consumption, X0 , at time 0 (X0 = A0 +w0 ) minus consumption at that time. Any resources that are not consumed earn a return 1 + r0 and can be used to finance consumption at time 1. Equation (8) is a difference equation in A which we can solve by assuming either initial or terminal conditions (see eg Chiang, 1984, or Sargent, 1987a). Assuming that AT = 0, that is, consumers leave no assets when they die and that interest rates are constant and equal to r, yields the following solution to (8): c0 +


1 1+r

c1 + · · · +


1 1+r


cT = A0 + w0 +


1 1+r

w1 + · · · +


1 1+r




This equation says that a consumer’s lifetime resources (given by the right-hand side of (9)) must be spent on lifetime consumption (the left-hand side of (9)). In standard constrained maximisation problems (such as that one represented by equations (6) and (9)) the solution is found at the point where the objective function (the function we wish to maximise or minimise) is tangent to the constraint function. In Appendix C we show with the aid of diagrams the solution to a two-period consumption problem. For a two period consumption problem, the solution is found where the slope of the indifference curve is equal to the slope of the budget constraint. An example will make things clear. Assume that consumers live for two periods, so that the two period equivalent problem of (6) and (9) is to maximise 1 u(c1 ) U (c0 , c1 ) = u(c0 ) + (1 + δ) subject to 1 1 c1 = A0 + w0 + w1 1+r 1+r where it is was assumed that agents do not want to leave bequests at time 2. The maximisation is obtained at the point where the indifference curve is tangent to the budget constraint. The slope of the indifference curve is c0 +

dc1 = − dc0

∂U (c0 ,c1 ) ∂c1 ∂U (c0 ,c1 ) ∂c0


u0 (c1 ) (1 + δ) u0 (c0 )

We now need to find the slope of the budget constraint. Writing the budget constraint as c1 = −(1 + r)c0 + (1 + r)A0 + (1 + r)w0 + w1 We see that the slope of the constraint is given by −(1 + r). Equating the slope of the indifference curve to the slope of the budget constraint yields −

u0 (c1 ) = −(1 + r) (1 + δ) u0 (c0 )

which is the Euler equation reported in the text, (12). 8


Rational expectations, unit roots and poor forecasting

Despite their early success, three related factors led to the demise of the LCH and PIH to explain aggregate consumption. The first factor was purely empirical. Consumption equations based on the PIH or LCH began to underpredict consumption, and the previous stable relationship between (current) consumption and (current) income was no longer found in the data. These empirical failures were the result of the impact of the cyclical components in economic variables which gathered momentum as the underlying economic environment became more volatile in the 1970s. The second factor was an econometric one. As Deaton (1992, p. 79) states: ‘It is a sobering undertaking to look back at many of the macroeconomic models of the time, and note the (now) obvious time-series problems: spurious correlations between integrated regressors, high coefficients of determination coupled with low Durbin-Watson statistics, and an almost complete lack of diagnostic testing.’ A revolutionary paper by Davidson et al. (1978) dealt with many of these econometric issues and initiated the development of a conventional methodology for empirical modelling that led to the formalisation of the now standard procedures of cointegration analysis, dynamic models and error correction. These concepts have been crucial; cointegration and error correction analysis have allowed economists to establish a clear distinction between long-run and short-run (dynamic) statistical relationships between economic variables which in turn have led to considerable insights into the relationship of consumption with variables that are thought to influence it both in the long and the short-runs. For instance, various empirical studies have noted that a stationary, or an equilibrium long-run, relationship between current consumption and current income is unlikely to hold since the secular, or trending components of these variables tend to exhibit significant divergence. To achieve the statistical stationary condition which describes the long-run behaviour of consumption, one needs to assume that consumption depends on other ‘secondary variables’ besides income in the steady-state. Personal wealth, relative prices, measures of income or age distribution, etc. appear to perform this secondary role successfully.14,15 The third factor evolved from the rational expectations revolution led by Lucas (1976). The Lucas critique states that in the face of rational expectations, structural relationships between variables may not exist. Lucas used the consumption function as an example; under rational expectations agents should only perceive a structural relationship between permanent income and consumption. But LCH and PIH also assert that a further structural relation between observed income and permanent income exists so that consumption is eventually determined by observed current income. Lucas argued that there was no reason to expect a stable relation between current and permanent income because changes elsewhere in the economy could alter the optimal way consumers make inferences about permanent income from observed income. Consumption depends on current and expected future incomes. The relationship between past and expected future incomes cannot be properly treated as an invariant feature of the economic environment and it is likely to change whenever changes in policy or other events cause rational agents to change the way in which past incomes affect forecasts of future incomes. What does not change is the structural relationship between consumption and permanent income. We can think of at least two implications for policy analysis that arise from the above arguments. First, the time series properties of the variables thought to influence consumption must be taken seriously for econometric work. Second, the theoretical and empirical implications of rational expectations must be considered. 14

See for instance, Hendry and von Ungern-Sternberg (1981), Borooah and Sharpe (1986). See section 11 in Muellbauer and Lattimore (1994, pp. 276-89) for some current examples. 15 This result has a relatively important policy implication since it suggests that when consumption depends on another variable besides income, a change in the income process cannot guarantee a corresponding change in consumption unless the other variable is entirely unaffected by the underlying policy.



The rational expectations permanent income hypothesis

Hall (1978) addressed the Lucas’ critique for consumption and in doing so also solved the time series problem of non-stationarity. He argued that the structural relationship for consumption did not emanate from the relationship between current consumption and current income but from the ordering of intertemporal preferences. What does not change in the face of expectations is the agent’s overall aim to maximise lifetime utility. The concepts and measurement of expectations and wealth, contrary to previously held beliefs that placed them in a second order, came to play a central role. Hall considered a permanent income model under uncertainty. Households choose a stochastic consumption plan to maximise the expected value of their time-additive utility function subject to an ‘evolution of assets’ budget constraint max V (ct , ct+1 , . . . , ct+T ) = Et

T −t X

(1 + δ)−τ u (ct+τ )


τ =0

subject to T −t X τ =0

R−τ (ct+τ − wt+τ ) = At


Et denotes the mathematical expectations operator conditional on information available at time t, δ is the rate of subjective time preference, R = 1 + r is the rate of return and r is the rate of interest which is assumed to be constant over time, c is consumption, A are assets apart from human capital, T is the length of economic life; u(·) is the one period-period utility function that is assumed strictly concave and time separable, and w are earnings which are stochastic and the only source of uncertainty. Equation (10) states that consumers’ lifetime utility is made up of the sum of (discounted) period-utility functions. Equation (11) states that lifetime consumption expenditure must be financed out of lifetime wealth (the sum of lifetime earnings plus initial assets). The first order condition for (10) and (11) yields the Euler equation16 ¸ · 1+δ 0 0 u (ct ) Et u (ct+1 ) = (12) 1+r Equation (12) states that the utility lost from giving up a unit of consumption – the right-hand side – must be equal to the expected utility gained by consuming the proceeds of the extra saving at any future date. We can approximate (12) by taking a first-order Taylor-series expansion around u0 (ct ): ¶ · ¸ µ ct+1 − ct 1 1+δ (13) −1 ≈ Et ct rav(ct )ct 1 + r where

1 rav(ct )


u0 (ct ) u00 (ct )

< 0. Equation (13) gives the ‘expression’ of consumption growth. To satisfy

the Euler equation, consumption will be growing, ct+1ct−ct > 0, when the interest rate is greater than the rate of time-preference r > δ, and declining when δ > r. The impact of changes to the growth rate of consumption is also governed by rav(c1 t )ct , the elasticity of substitution for consumption. For (12) to hold, agents must choose consumption optimally each period given all the available information at the time the decision is made. Consider a reduction in consumption at time t 16

For the mathematics behind this result, see Appendix A. Box C shows the assumptions required to derive the Euler equation. Note that (12) is an equilibrium relationship and not a consumption function. It is also used extensively outside the consumption literature; for example, the Capital Asset Pricing Model (CAPM) is derived from it under the assumption of stochastic returns; the standard intertemporal transmission mechanism in Dynamic Stochastic General Equilibrium (DSGE) models is a log-linear version.


from the value the individual had chosen to satisfy the Euler equation to finance an increase in consumption in the future. A marginal change of this type should not increase lifetime utility; otherwise the previously thought optimal choice ct would not be optimal since it would not yield the maximum amount of lifetime utility. When utility is quadratic, the Euler equation is ¸ · 1+δ ct = Et ct+1 (14) 1+r Repeated substitution of (14) into (11), assuming r = δ yields a consumption function T −t X ª © At + ct = ytp = λt × Wt = PT −t (1 + r)−τ Et (wt+τ ) τ =0 (1 + r) τ =0


ytp denotes permanent income which is defined as the proportion, λt , of expected lifetime wealth, Wt . Lifetime wealth is, in turn, defined as the sum of non-human and human wealth. Consumption is a linear function of initial wealth and the present value of expected future income. Higher moments of income do not matter, only its mean. The following equation summarises many of the points made above: ct+1 = ct + εt+1 .


The error term, εt+1 , is unpredictable at t. Statistically speaking, (16) can approximate the stochastic behaviour of consumption closely since for many countries consumption is a random walk.17 As such, (16) states that the best forecast about the level of consumption in the next period is today’s level of consumption! Differences between consumption at t and t + 1 are brought about by unpredictable events at time t+1; these are summarised by εt+1 . Although the change in consumption is unpredictable, (16) satisfies the rational expectations premise. A rational expectation will use all available information at the time the expectation is formed. Given the information available at time t, agents will set consumption, ct , equal to their estimate of their permanent income. If no information about the future has become available between t and t + 1, in t + 1 the consumer’s estimate of his or her permanent income will be unchanged so ct+1 will be equal to ct . Only if new information becomes available between t and t + 1 will consumption change in t + 1. As new information is unpredictable by definition, it must be the case that consumption differs from lagged consumption only by an unpredictable element. Hence, εt+1 conveys information about the impact of all new information that becomes available to consumers in t + 1. All past/predictable information is reflected in the lagged consumption term. Thus REPI predicts that no lagged variable other than ct can predict ct+1 ; all lagged information is embodied in ct and should not affect ct+1 . What are the implications of this framework for policy makers? Box B explains the impact that changes in interest rates have on consumption. The box shows that increases to the interest rate will increase the growth rate of consumption.18 However, the theory is not able to determine the impact that increases in the rate of interest have on the level of consumption; the overall impact will be negative if and only if the sum of the substitution and wealth revaluation effects outweigh the income effect. Elmendorf (1996) and Cromb and Fernandez-Corugedo (2004) show for that for most of the combinations of standard parameter values, the overall impact of an increase in interest rates on consumption will be negative. There are other important implications of this framework for consumption. REPI stresses the importance of rational forward-looking consumers. Under rational expectations, expected changes 17 Equation (16) does not say anything about the variance of ε, and there is no reason to believe that the variance is constant. Hence, strictly speaking, equation (16) is not a random-walk (Deaton, 1992). 18 In order to satisfy the Euler equation for the set of preferences in Box B.


to (after-tax) labour income, or interest rates any time in the future will influence consumption today and should not affect consumption in the future. For instance, if the government were to announce a (permanent) decrease/increase in taxes taking effect at some point in the future, and consumers were to believe it, consumers would immediately increase/decrease their consumption at the time the policy announcement was made and not at the time the policy is implemented. A similar interpretation could be given to pre-announced changes to interest rates: a central bank’s pre-announcement that interest rates will change at some point in the future would lead to a change in consumption at the point the announcement was made. However, a non-trivial problem must be noted. The fact that permanent income and expected lifetime income or wealth are not directly observable is a major handicap in carrying out empirical work that is consistent with the theory; see (15). Thus for the purposes of empirical and forecasting work, a measure of permanent income is required. But such a measure has proved elusive and only crude proxies have been used. Thus, it is not surprising to find that many empirical economists cannot find evidence in favour of REPI.

Box B: The impact of interest rates on consumption In this box we examine the effect that interest rates have on consumption. c1−θ

t where θ determines the Assume that the utility function is of the CRRA case, u(ct ) = 1−θ elasticity of substitution between periods, see Box A. The Euler equation is

· ¸1 1+r θ ct+1 = ct 1+δ To satisfy the Euler equation, an increase in the interest rate, r, must be matched by higher consumption growth. This increase is scaled by both θ and δ. The level of consumption is ¶i 1 ( ct = λ · Wt , λ = 1+r i=0 | {z } T −t X


from constraint



1+r 1+δ | {z





from Euler equation

λ is the marginal propensity to consume (mpc) out of lifetime wealth, W . The impact of a change in interest rates on consumption is given by ∂λ ∂Wt ∂ct = Wt + λ ∂r ∂r ∂r } | {z } | {z Inc+sub


The change in the mpc encompasses the income and substitution effects. The second term captures the wealth effect. An increase in the interest rate, makes consumption at t more expensive compared to consumption thereafter. With a convex indifference curve for consumption, agents will want to postpone consumption (the substitution effect, found in the Euler equation), but with cheaper consumption in the future, consumers can afford more current consumption (the income effect, found in the constraint). The wealth effect occurs because with higher interest rates, the future is discounted more and the present value of lifetime resources decreases. Thus the overall effect on the level of consumption is unknown. See Appendix C for more.


Box C: The role of assumptions in REPI Assumptions must be made to obtain a solution to the consumption problem when there is uncertainty.a For REPI we have: i) Consumption is the only argument in the consumer’s utility function; ii) Capital markets are perfect so that consumers can borrow/lend without any restrictions at a constant rate as long as the present value of their consumption does not exceed the present value of their human and financial wealth; iii) The rate of time preference does not exceed the rate of interest; iv) Existence of certainty equivalence; v) Agents are all equal.b The first two assumptions allows substitution between current and future expenditures to achieve the maximum level of lifetime utility. The ability to borrow and lend makes the optimal consumption plan independent of current income under certainty (note that current income affects consumption plans as income affects consumption through its unpredictability - the error term in the consumption equation). This explains why consumption plans are independent of the level of current income and only depend on preferences, interest rates and unforeseeable events. The third assumption restrains impatient consumers and prevents them from going into substantial debt. The fourth assumption allows an analytical solution to the consumption function. Deviations from this assumption make it difficult, if not impossible, to obtain analytical expressions for the level of consumption. The final assumption makes it easy to aggregate. a

In the face of uncertainty, analytical solutions are possible if:

1. The utility function is quadratic 2. The utility function is of the constant absolute risk aversion (CARA) and the only risk is to labour income 3. The utility function is of the constant relative risk aversion (CRRA) and the only risk is to the interest rate. There are no solutions when the utility is CARA and the risk is to the interest rate, or if utility is CRRA and the risk is to labour income. For a mathematical exposition to these facts see Carroll and Kimball (1996). b Other implicit assumptions are: vi) no adjustment costs, vii) non-durable goods assumed only, viii) no measurement errors or transitory shocks to consumption, ix) the coincidence of the frequency of consumers’ decision making with the observation period of the data, x) infinite lifetimes and xi) rational expectations.


Departing from REPI

Despite REPI’s prediction that consumption is a random walk, and the statistical finding that consumption in most countries is I(1), two results refute REPI:1920 1. Consumption reacts too strongly to predictable changes in actual income. This is the socalled ‘excess sensitivity’ phenomenon pointed out by Flavin (1981). If expectations are rational then Hall’s REPI should be refuted since the claim that changes in consumption are 19

See Deaton (1992, ch. 3 and 4) for more details. Attanasio and Weber (1993) have argued that departures from REPI occur at the aggregate level due to aggregation issues. Evidence on the micro/panel data is less conclusive; see Deaton (1992) chapter 5 for evidence in favour and against REPI. Aggregration issues are considered in this section and the next. 20


unpredictable is not fulfilled (past income is able to predict consumption).21 2. Consumption reacts too weakly to changes in permanent income. This so-called ‘excess smoothness’ phenomena was pointed out by Deaton (1987). This result is particularly damaging to REPI since it suggests that permanent income is more volatile than consumption thereby defying the original purpose of the permanent income hypothesis which attempted to explain why consumption appeared to be smoother than actual income. Almost all theoretical consumption papers after Flavin (1981) and Deaton (1987) attempt to explain both phenomena.22 Below we list a number of influential theoretical candidates (most of which have been derived with the relaxation of some of the assumptions made in REPI (see Box C)). The implications for policy makers are discussed. Since the theories that follow are all capable of explaining excess sensitivity and smoothness, an initial (and simple) test for the validity of these theories in different countries will be to test for the existence of excess sensitivity and smoothness. Appendix B shows how to test for excess sensitivity and smoothness.


Breaking certainty equivalence: precautionary saving

In the previous section we saw that when utility is quadratic only first moments (ie means) affect consumption, see (15). Second or higher moments play no role. Many authors have argued that second moments are important for consumption behaviour (Leland (1968), Sadmo (1970), Kimball (1990), etc.). To see this, take a second-order Taylor-series expansion of (12) around u0 (ct ): ¶ µ · ¸ µ ¶µ ¶ h i ct+1 − ct 1 1+δ 1 Ψ(ct ) −1 + (17) ≈ Et (ct+1 − ct )2 Et ct rav(ct )ct 1+r 2 ct | {z } | {z } precautionary saving



(ct ) Ψ(ct ) = − uu00 (c . The growth rate of consumption now depends on two components; the first comt) ponent we saw in (13). The interesting term is the second term which we have called ‘precautionary saving’. With convex marginal utility, u000 > 0, the second term is positive and the growth rate of consumption is higher compared to the case where utility is quadratic (where u000 = 0). Why is this important? The higher the growth rate of consumption, the lower the level of consumption must h be if (11) iis to be satisfied. Thus the larger the variance of the growth rate of consumption, Et (ct+1 − ct )2 , the higher the growth rate of consumption and the lower the level of consumph i tion.23 But Et (ct+1 − ct )2 is not the only relevant term; Ψ(ct ) acts as a scaling factor that determines the strength of precautionary saving (Box D). To see how precautionary saving affects consumption, assume utility is of the Constant Absolute Risk Aversion (CARA) form, u(ct ) = − γ1 exp [−γct ], (Caballero, 1990a). If r = δ and labour income follows a random walk with normally distributed innovations that have standard deviation σ w , the level of consumption is given by γσ w (18) ct = ytp − 2 where is ytp defined as before. The difference between (15) and (18) is γσ2w which represents precautionary saving. This term is determined by uncertainty, σ w , and prudence, Ψ(ct ) = γ; higher 21 The predictability component depends on the nature of the income process so that if income follows a unit root the predictable component carries over to the future. 22 These two findings have also been found in the UK, France, Canada, Japan and Sweden (Campbell and Mankiw (1989, 1991)). Japan is found to be borderline on excess smoothness. Jappelli and Pagano (1989) also find excess sensitivity in Sweden, the US, UK, Italy, Spain and Greece; Arreaza (2003) in Latin American countries. 23 This extra saving is termed precautionary saving. If Ψ0 < 0, we have decreasing absolute prudence.


uncertainty or more prudence leads to a lower level of consumption and higher precautionary saving. (Note that higher uncertainty and/or prudence leads to higher consumption growth.) Caballero (1990a) argues that (18) can explain many of the time series properties of US aggregate consumption including excess sensitivity and smoothness. However, CARA is not used much in theoretical consumption papers. This is because it does not rule out negative consumption and it implies that wealthy agents are as risk averse and prudent to small risk as poorer agents (ie they have constant absolute risk aversion and constant absolute prudence – see Box D for more). There are important implications for policy makers arising from precautionary saving. A model with precautionary saving highlights the importance of uncertainty and volatility. The more uncertain or volatile the economy is the higher precautionary saving and the lower consumption are. Thus, central banks and/or governments which aim to stabilise the economy (ie try to reduce uncertainty) are likely to lower precautionary saving and therefore increase consumption, all else equal. For instance, precautionary saving has been given as an explanation for the US and UK consumption booms of the 1990s. For both countries the economic environment was less volatile during the 1990s than it had been the previous two decades.24 This led to a fall in precautionary saving (the US and UK saving ratios were at all time lows in 1999) and to a boom in consumption expenditure. Other authors have claimed that part of the explanation for the sluggishness of the Japanese economy during the 1990s was due to the high amount of precautionary saving. 5.1.1

The buffer stock model

Precautionary saving is thought to be an important determinant of consumption. The buffer stock model is a stylised ‘extension’ to the framework of precautionary saving under specific assumptions made to match both micro and macro data (Deaton, 1991, Carroll, 1992, 1997). In particular, Carroll has argued that the buffer stock model is able to explain the recession of the early 1990s in the US as well as the subsequent boom in the 1990s. The arguments that follow are taken from Carroll (1992, 1997, 2001). The buffer stock model introduces very specific attitudes towards risk. It assumes decreasing risk aversion, prudence and decreasing prudence. The conjecture that labour income has a small chance of falling to zero, depicting the small possibility that the consumer becomes unemployed is added to the assumptions made about risk.25 A final condition, the ‘impatience condition’, is imposed. This condition states that if no risks exist, then individuals would want to consume more than their current income: (19) (βR)1/θ < G where G = 1 + g denotes the growth rate of labour income, β = (1 + δ)−1 and θ is the coefficient of relative risk aversion in the CRRA utility function.26 We have all the tools needed to explain buffer stock behaviour. Uncertainty makes individuals want to (precautionary) save to be able to meet future uncertainty. This entices consumers to accumulate large levels of wealth, all else equal. But the impatience condition restricts this accumulation. These two opposing forces lead consumers to engage in buffer stock saving behaviour: consumers form a target level of wealth holdings such that, if current wealth is above target, the impatience condition dominates and consumers increase their consumption to reach the target, whereas for levels of wealth below target, prudence dominates and consumers choose to add to their 24

See Stock and Watson (2003) for a recent paper arguing that the world has become less volatile in the 1990s compared to the 1970s and 1980s. 25 Instead of zero, the model can be modified so that labour income falls to a lower band which proxies unemployment benefits. See below for more. 26 The left hand side of the equation is approximately equal to the growth rate of consumption and the right hand side to the growth rate of labour income. To see this, take a log approximation of this expression to give θ−1 (r−δ) < g.


wealth to reach the target. The stronger the impatience (prudence) condition, the lower (higher) the target level of wealth. To understand how the model’s parameters affect consumption one only needs to understand their impact on both the prudence and impatience conditions. As an example, consider an increase in g, the growth rate of labour income. Such an increase increases the strength of the impatience condition. This, in turn, leads to a lower level of target wealth and therefore higher consumption. A decrease to the discount factor δ, on the other hand, decreases the impatience condition leading to an increase to the target level of wealth. An increase in uncertainty increases the prudence condition thereby increasing the target level of wealth. The Euler equation for the buffer stock model is: µ ¶ θ vart (∆ ln ct+1 ) (20) Et ∆ ln ct+1 ≈ θ−1 (r − δ) + εt+1 + | {z } 2 CEQ

if shocks to consumption are lognormally distributed. Consumption growth depends on three factors; the degree of impatience over precaution, a random effect and the conditional variance of next year’s consumption given information available this year. The first two components are standard to intertemporal consumption behaviour and have been termed CEQ. The variance term is proven to play a significant role in consumers’ behaviour, a characteristic of the buffer stock model. The average (aggregate) variance of consumption is given by µ ¶· ¸ σ 2ln N 2 −1 g− − θ (r − δ) (21) Et [vari,t (∆ ln ci,t+1 )] ≈ θ 2

where σ 2ln N denotes the variance of (permanent) labour income and we have taken the expectation across all households as of time t. Thus, the variance of consumption growth increases with the growth rate of labour income, g and the rate of time preference, δ. It falls when uncertainty increases (ie when σ 2ln N increases). The coefficient of relative risk aversion, θ, has offsetting effects; a higher coefficient represents a stronger precautionary motive (more wealth accumulated due to the 2θ term) but at the same time, a higher θ leads to a lower intertemporal elasticity of substitution: thus θ−1 (r − δ) decreases, increasing the impatiency condition. It is important to note that the variance of consumption growth is an endogenous term. It is also worth pointing out that if consumption represents a large fraction of GDP, then movements in the variance of consumption growth are likely to affect movements in GDP growth too.27 The buffer stock model has a number of interesting and relevant implications for policy makers. Its predictions emanate from the role that wealth plays in the model; agents form a target level of wealth which they try to match at all times. It provides a clear role for many of the variables of the model through their impact on the target level of wealth. Unlike REPI, it predicts a clear negative role for interest rates on the level of consumption. The higher the rate of interest, the higher the target level of wealth (and the lower the variance of consumption) and the lower the level of consumption. The model gives an important role to uncertainty. The higher uncertainty is, the greater the target level of wealth and therefore the lower the level of consumption will be. It obviously provides a clear role for balance sheets; if agents do not have sufficient wealth to buffer themselves against uncertainty, they will save more to build up their stock of wealth and therefore lower consumption. There is also a role for expected labour income growth. If agents expect their labour income to grow faster in the future, they will lower their target level of wealth and consume more. 27

Obviously, to consider linkages between the growth rate of consumption and the growth rate of GDP, one would need a general equilibrium model.


An interesting implication of the buffer stock model is that the marginal propensity to consume out of permanent income shocks is less than one (although not very different from one), unlike REPI’s predictions.28 The implications for policy makers are crucial; an announcement that taxes will be permanently reduced by x% will not lead to an equivalent x% increase in consumption. A final interesting implication for policy makers is the speed to which agents correct changes from the buffer level of wealth. Whilst no analytical solution exists in this type of models, numerical simulations suggest that adjustment to the target level of wealth is fairly rapid.


Near rationality and aggregation

A number of authors have argued that the reason consumption is close to being described by REPI but suffers from excess sensitivity and smoothness, is because agents are not as rational as economists think. Agents do not have time to solve complex dynamic programmes like those shown in Appendix A.29 Instead they follow simple rules that enable them to get close to the optimal solution. Akerlof and Yellen (1985) have argued that if agents are close to the optimum (of say the level of consumption), shocks which take them away from this optimum may lead them to incur costs (of adjusting) that exceed the benefit from moving back to the optimal point. Thus, it is possible that following simple non-optimal rules when consumption is close to the optimum may not lead to large losses of lifetime utility. For example, Cochrane (1989) looks at the utility loss which agents incur when they follow alternative (not rational) decision rules instead of REPI. He finds that the utility cost to an agent that decides to set consumption equal to current income rather than to permanent income is between ten cents to a dollar per quarter. Intuitively, the utility costs are small because the utility costs of deviating from an optimum are an order of magnitude smaller than the deviation itself. In this section we shall examine two sources of near-rationality; those originating from a failure of agents to update their consumption decisions optimally every period and those originating from agents’ decision to not use all available information when forming optimal consumption decisions. 5.2.1

Near rationality and the updating of consumption decisions

Allen and Carroll (2001) examine whether agents can, by trial and error, approximate a nearrational rule that leads to small utility loses compared to those obtained by solving the complex dynamic programming problem that leads to the buffer stock model. They argue that agents should be able to undertake this approximation which comprises two steps. In the first step, agents determine a target level wealth, X ∗ , and a parameter that will tell them how quickly to return to X ∗ after facing a shock that moves them away from X ∗ . In the second step, once X ∗ is achieved, agents equate the growth rate of consumption to the growth rate of income. Allen and Carroll find that this rule, whilst intuitive, simple, and consistent with Cochrane’s findings, takes a long time to get right (even though this is not the optimal solution to the problem). Thus, agents may not always choose near-rational alternatives since these are themselves difficult to complete. Nonetheless, Allen and Carroll argue that all is not lost because agents may learn from each other, so that learning about simple rules is only possible if agents interact. 28

This point is made eloquently by Carroll (2001). For REPI models, given the budget constraint, a permanent increase to income today (and every subsequent period) will lead to a permanent increase in consumption (in the budget constraint (11) both sides have to be equal). For the case of the buffer stock model, the mechanism is a little more elaborated. A permanent shock to labour income, raises permanent income and the ratio of wealth to permanent income is somewhat depressed. Consumers therefore have to build this ratio up and not all of the shock is consumed immediately. Hence the result that the marginal propensity to consume is less than one. 29 Agents may not know mathematics, or how to use computer software which may allow them to solve these models. They may not know the exact form of the utility function!


Agents probably do not solve complex mathematical problems but instead follow simple rules that may lead to small utility losses. If near-rationality is important for individual consumption, what does it imply for aggregate consumption? It may be tempting to conclude that as small deviations from the optimum do not lead to large effects on individual consumption, then aggregate consumption would not be affected much by such small deviations. However, this is far from the truth. Akerlof and Yellen (1985) showed that whilst consumers may incur small (second-order) losses in utility by following rules of thumb, if there is a sufficiently large number of agents that follow these rules, the overall effect on the economy may be large (first-order). Caballero (1995) examines ‘whether microeconomic near-rationality in the Akerlof-Yellen (1985) sense has the potential to generate aggregate consumption dynamics similar to those observed in actual U.S. data’ [p. 30]. The model works as follows. Assume a large number of consumers. If agents update their consumption patterns continuously and are rational, their consumption would follow (15); in logs c∗i (t) = li + wi (t), where letters with an i subscript refer to individual i. If the economy grows, c∗i (t) is approximated by a random walk with drift (consumption grows in the steady-state). In this economy agents are near-rational and decide to maintain a constant level of consumption, ci (t), whenever possible. However, if zi (t) ≡ ci (t) − c∗i (t) – the difference between optimal consumption and the level of consumption of the near-rational consumer – reaches a lower (L) or an upper (U ) trigger point, consumption is set equal to the optimal rule, ie ci (t) = c∗i (t).30 The growth rate of consumption for near-rational agents is dci (t) = dzi (t) + dc∗i (t). Aggregate consumption growth is (22) dC (t) = dC ∗ (t) + dZ(t) with dZ(t) ≡



zdf (z, t) dz



where f (z, t)31 represents the cross-sectional density of zi s at time t. Upper case letters denote the the aggregate counterparts of the lower case, individual, variables. As REPI is assumed, aggregation from c∗ to C ∗ is trivial (it is equal to the average of all individual optimal consumption functions, since these are all equal).32 Aggregation of the individual z 0 s, which depends on f (z, t), drives the dynamics of aggregate consumption, C. Thus, the cross sectional distribution of agents in the economy, f (z, t), is crucial for explaining aggregate consumption. It is assumed that if the economy has not suffered any shocks for a number of periods the cross sectional distribution f (z, t) should be at or close to the steady-state distribution.33 What happens if the economy suffers a number of positive and constant aggregate shocks? What are the implications for policy makers? In the absence of near-rationality, rational consumers update their REPI level of consumption to match the size of the shocks. In that case aggregate consumption changes accordingly. With near-rationality the actual rate of aggregate consumption growth only picks up slowly. This is because not all near-rational agents reach a trigger point at the same time so that consumption for many does not change. As a result, the cross sectional distribution also changes slowly as the economy slowly adjusts its the steady-state cross sectional distribution. Sluggish aggregate consumption dynamics result. 30

These type of models are commonly called (S,s) models/rules. The notation f (·) used throughout indicates that the function or distribution in question is a function of the arguments inside the brackets. 32 Thus the aggregate growth rate of consumption for rational consumers, dC ∗ (t), can be approximated by a drift term. 33 The steady-state cross sectional distribution is skewed to the right. This is due to the assumption of a growing economy which implies that there is a higher percentage of consumers upgrading their consumption patterns than downgrading them, ie more consumers are reaching the lower trigger L than the upper trigger U. 31


This framework suggests that the cross sectional distribution of agents in the economy provides a rich source of information for consumption. As Caballero notes, ‘the magnitude and timing of the response of consumption to wealth innovations depend on the shape of the cross-sectional density at each point in time, which depends on the stochastic environment faced by consumers and on the path of aggregate shocks in particular.’ [p. 35.] Obviously, it is almost impossible to observe cross sectional distributions in reality. Nonetheless, it may be possible to use micro data sets, surveys, or other sources of data to provide proxies for cross sectional densities. These proxies may provide information that can aid forecasts of future aggregate consumption expenditure. Another important prediction of the model is that if, in the recent past, the economy has experienced a series of shocks in the same direction, it is more likely than not, that consumers would have had adjusted their behaviour recently, suggesting that another shock is unlikely to make them reach a trigger point that will make them re-adjust. Thus, we may think that following a series of positively correlated shocks (say an expansion or a recession), consumers are less likely to respond to future shocks. Another implication emanates from these arguments; consumption responses are likely to be asymmetrical and the use of non-linear time series econometrics may do a better job in explaining consumption behaviour at the aggregate level.34 5.2.2

Near rationality and imperfect information

Another source of near-rationality is imperfect information. Pischke (1995) explores the implications for an economy where agents do not always update their information sets optimally, choosing to ignore important information in key economic variables. If a sufficiently large number of agents behave in this manner there can be important implications for aggregate consumption. We consider the simplest version of Pischke’s model in the arguments that follow. Assume that agents are rational and follow REPI, but aggregate information plays ‘little role in household decisions since the economic environment in which individuals operate differs sharply from the economy as it is described by aggregate data.’ [p. 806]. To solve the model analytically it is assumed that all individuals have identical labour income processes, but each agent faces a different realisation of that process every time period. For ease of exposition we take the simplest income process which consists of a random walk with innovations that are common to all individuals and a white noise component with shocks that are uncorrelated across individuals (24) ∆wit = εt + uit − uit−1 where i subscripts denote individual variables and no subscripts refer to aggregate variables. Note that the aggregate shocks will have a permanent effect on labour income but that the individual ones only a transitory impact. It is assumed that individual shocks (the ui ’s) are mutually uncorrelated and will sum to zero for a large population n 1X (25) ∆wit = εt ∆wt = n which is the per capita income process a time series analyst observes from the aggregate data and n is the size of the population in this economy. Because consumers cannot distinguish between aggregate and individual income components, ie the agent cannot distinguish between εt and uit , their income process takes this form ∆wit = η it − θη it−1 = (1 − θL)η it

f (σ 2u ,

σ 2ε ),

σ 2ε

σ 2u


where η it = f (εt , uit ), θ = and denote the variances of the aggregate and individual errors respectively and L denotes the lag operator.35 Given (15) and (26), the change in 34 35

See section on aggregation below for more on the implications of non-linearity and aggregate consumption. Hamilton (1994, pp. 102-8), shows how to derive appropriate expressions for θ and η it .


individual consumption is given by36 ∆cit = (1 − θρ) η it ≡ Aη it = A

∆wit 1 − θL


the last equality comes from (26) and the invertibility of 1 − θL. Using (25) we can obtain the aggregate consumption change as n

∆ct =


∆wt εt 1X 1 X ∆wit =A =A ∆cit = A n n 1 − θL 1 − θL 1 − θL


Consumption innovations follow an AR(1) process, ∆ct = Aεt +θ∆ct−1 . Thus, changes in aggregate consumption are sluggish.37 There are a number of implications arising from these simple theories. Theories of near-rationality suggest that for a given shock (say a policy announcement) innovations in consumption are more sluggish than the equivalent response suggested by REPI. Despite this prediction, consumers are still forward-looking so many of the predictions of REPI discussed earlier still hold. A final point to note is that consumption adjustments following shocks may be asymmetrical so that a reduction in interest rates may have a different impact on consumption than an equivalent increase.38


Liquidity constraints

Consumers’ inability to borrow (or liquidity constraints) are thought to be an important reason for the failure of REPI. The literature normally considers two scenarios: one where consumers may face high borrowing costs (soft liquidity constraints) and another one where consumers are not allowed to borrow at all (hard liquidity constraints).39 Both scenarios lead to lower consumption compared to REPI if agents have low resources at their disposal. Current income has as a result a much more important role determining consumption behaviour than future income. A typical consumption problem with liquidity constraints comprises the maximisation of equations (10) and (11) plus a constraint that wealth never be negative: At ≥ 0 at all times


The implications of this problem for consumption can be examined through the Euler equation (Deaton, 1991): (30) u0 (ct ) = max[Et βu0 (ct+1 ) R, u0 (At + wt )] When the liquidity constraint binds, consumers (who would like to borrow but cannot do so) are forced to spend all of their current resources, At + wt . Since current resources are less than the 36

See Appendix B. Note however, that the innovations at the individual level satisfy REPI as they are a random walk. At the aggregate level they are sluggish. Thus, this framework is able to reconcile evidence which suggests that REPI can explain consumption at the individual level but not at the aggregate one. The model presented in section 5.5 also satisfies this premise. 38 This type of information mechanism has recently been advocated by Mankiw and Reis (2002) within a DSGE for policy analysis. 39 As Fernandez-Corugedo (2002) shows, the behaviour arising from hard and soft constraints is very similar. In this section we only present the arguments related to hard constraints. For more see Fernandez-Corugedo (2002) and references therein. 37


desired level of consumption, (ie At + wt < c∗t ), marginal utility will be u0 (At + wt ) when the constraint binds.40 If the constraint does not bite, then we have the standard Euler equation (12). What are the implications of (30) for consumption? If consumers cannot borrow, they will be forced to reduce their level of consumption (ie they will save more out of the same lifetime resources). Why do borrowing constraints lead to behaviour that resembles precautionary saving?41 Compared to the case where there are no borrowing restrictions, consumers undertake extra saving to avoid situations where their borrowing leaves them exposed to having few resources that can be used to finance future consumption. For instance, take an extreme situation where borrowing is not allowed if income falls to zero. If consumers were to borrow at t and then faced the possibility of having no income at t + 1, they would not be able to consume anything at t + 1 causing them a large drop in utility. As consumers wish to avoid events like these, they will never leave themselves exposed by having few assets. Thus, the introduction of liquidity constraints increases the precautionary saving motive (ie it increases prudence). Another way to think about these arguments is to note that the ability to borrow in bad times serves as insurance to some consumers when their income falls. Taking this form of insurance away from them is akin to increasing risk which leads to higher precautionary saving. There is another important implication arising from (30). Even when the borrowing constraint is not in effect, consumption is still be affected. Why? Even if consumers are not constrained today, the liquidity constraint may bind at some point in the future. This will affect the Euler equation ‘in the future’. But recall that the Euler equation relates consumption between periods in such a way that if future consumption is affected, current consumption will also be affected. In other words, as consumers are forward-looking, they will take into account the possibility of being constrained in the future. As a result, the level of consumption for those who cannot borrow at some point in the future will be lower compared to a consumer who does not face the possibility of being constrained during its lifetime. There are a number of implications of liquidity constraints for policy makers. When facing liquidity constraints, consumers’ time horizon may be shortened. This makes consumption more likely to be influenced by current resources than future ones.42 With borrowing restrictions consumption may be best explained by (1) than by (15) and interest rates may play little role in determining consumption. Why this lesser role for interest rates? Assume agents cannot borrow. The central bank decides to reduce or announce that interest rates will be reduced in the near future to stimulate the economy. If agents are not able to borrow to increase consumption following the interest rate decrease, the impact on consumption will be modest. The model also has strong implications for events such as financial liberalisation. With more financial liberalisation (and therefore less consumers being liquidity constrained) the level of consumption increases as the precautionary motive introduced by the liquidity constraint disappears. Moreover, as the relaxation of borrowing constraints leads to more agents’ in the economy behaving according to standard REPI or buffer stock models, financial liberalisation ought to increase the response of consumption to changes to the interest rate. 40

This is a direct consequence of the assumption of decreasing marginal utility (or a concave utility function). Given that marginal utility is decreasing in its argument and that At + wt < c∗t , then u0 (At + wt ) > u0 (c∗t ) making (30) reduce to u0 (ct ) = u0 (At + wt ) when the constraint binds. 41 When liquidity constraints bite the Euler equation is given by u0 (ct ) = u0 (At + wt ). Thus, when the constraint binds, the Euler equation tells us that u0 (At + wt ) > Et βu0 (ct+1 ) R, which in turn implies that the left hand side of the Euler equation, u0 (ct ), will be greater compared to the case where there are not liquidity constraints. With decreasing marginal utility, if u0 (ct ) is greater due to the liquidity constraint, then the level of consumption at time t must be lower. 42 When the constraint binds, the Euler equation is a function of current resources, At + wt .




Habits have been proposed as a potential explanation for a series of macroeconomic topics/puzzles that cannot be explained with standard time-separable utility functions: the equity premium (Abel, 1990, Constantinides, 1990, and Campbell and Cochrane, 1999), why high growth causes high saving and not vice versa (Carroll and Weil, 1994, Carroll et al. 2000), and why consumption appears to be too smooth at high frequencies (Furher, 2000). In this section we show how habits affect consumption behaviour. Throughout we use a utility form that has recently become popular, that of multiplicative habits u(c, z) =


¢ c 1−θ zγ



where z is the reference stock. The parameter γ determines the importance of z: if γ = 0 the utility function reverts to the standard CRRA utility function implying that consumers only care about the level of consumption; if γ = 1 then only consumption relative to the reference stock is all that matters. The reference stock is a weighted average of past consumption. It evolves according to the following equation (32) zt = zt−1 + λ(ct−1 − zt−1 ) where λ determines the relative weight of consumption at different times. The lower λ is the less important recent consumption becomes so that if λ = 0 then past values of z do not matter in (31) and habits play no role. The Euler equation in the case of habits (assuming no uncertainty) is given by (see Carroll, 2000a) ¤ ¢¤ £ ¡ £ (33) uct = Rβ β λuzt+2 − (1 − λ) uct+2 + uct+1 £ z ¤ z −β λut+1 − (1 − λ) ut+1

where uc = ∂u(c,z) and uz = ∂u(c,z) ∂c ∂z . If either λ = 0 or γ = 0, then the standard Euler equation for consumption results. What are the implications of (33)? Taking a first order Taylor-series approximation around a constant growth rate for consumption, ct+1 = σct , and around a constant habit stock to total consumption ratio, hctt = χ, we have43 (ct+1 − σct ) ct

(ct+2 − σct+1 ) ≈ Γ − ct+1


(ct+2 − σct+1 ) − ct+1


≈ Γ

zt ct ucc

1 ct ucc


Ω · uc c ucc | t{z }


Like (13)

[zt + Ω · uc ]


44 What does (34) tell us about consumption where Γ, Ω, and zt are functions of σ, χ, R, ucz t and β. behaviour? First, note that (34) suggests that consumers now smooth not only the level of consumption (the left-hand side of (34)), but also its growth rate (the first term in (34)). This makes sense, the terms inside the utility function are both the level and the growth rate of consumption. It is important to note that the growth rate of consumption is next period’s; in other words, with the habit specification used here, agents are forward-looking and consider not only the impact of their decisions on the future level of consumption but also on the future growth rate. Moreover, 43 In this section we seek to explain the role that habits play, not so much the implications of uncertainty, hence the first order Taylor-series expansion. 44 For a full derivation see Appendix D. In the previous approximations, the growth rate of consumption was assumed constant, that is σ = 1.


note that given that Γ > 0 and Ω ≥ 0 then if z > 0 the growth rate of consumption for the case of habits is greater than the growth rate of consumption in the case of no habits. What are the implications of habits for policy makers? The habits framework has implications for the way in which agents respond to changes in policy instruments. Habits make consumption react more sluggishly to policy announcements, delaying the full impact of policy decisions.4546 Thus, following an announcement that interest rates will change, consumers will not respond as much as the case where there are no habits. A number of interesting implications arise if habits are introduced in a buffer stock model of consumption. Carroll (2000c) shows some of the policy implications that arise from a buffer stock model that has habits. One of the implications is that the marginal propensity to consume is much lower than in the case where there are no habits. According to Carroll the mpc can be as low as 30%. Hence policy makers wishing to increase consumption with (permanent) tax reductions will find that consumption may not respond much to a (permanent) tax change. Moreover, the adjustment that individuals make to deviations from the target level of wealth will be more sluggish than in the case where there are no habits.


Finite lives and REPI

Clarida (1991) studied the aggregate stochastic implications of Modigliani’s life cycle hypothesis (when there are rational expectations). Clarida asks whether a lifecycle model with rational expectations can explain the first and second moment properties of changes in per capita consumption. A key finding of the paper is that because saving is required to finance consumption in retirement, agents will not react so strongly to permanent/persistent changes in their labour income. This means that the mpc of an innovation to permanent income is less than one (REPI assumes an mpc of one47 ) and declines monotonically with age. This and other findings can be summarised in terms of four expressions. Aggregate consumption per capita is given by48 ∆ct = ϕλ + µet + φη t−1 where µ=


µ (j) ; µ (j) =


1 + R−1 + · · · + R−( −j) <1 1 + R−1 + · · · + R−(n−j)

ϕ≡ and η t−1


¶¸ · µ n−1 −1 µj + 1 1 X 1 X = εt−n+i + εt−n+j 1 − n n µ (1) i= +1


(37) (38)



n is the number of periods the individual lives, is the number of periods the individual works for (n − is therefore the retirement period), j is the age ¡ of a consumer ¢ ¡ at time ¢ t, µ is the marginal propensity to consume out of labour income and et ≡ nεt and λ ≡ ng are functions of the error and the drift term in the following specification for labour income wt = g + wt−1 + εt . The following implications come from (36)-(39): 45

Interested readers can follow Carroll (2000a) for the dynamic difference equations for consumption growth and the habit stock. These determine how quickly consumption is to approach the steady-state levels of consumption growth and the habit stock. 46 Many general equilibrium small macro models have started to introduce habit formation to the utility function. This is because the impulse responses for consumption generated by habit utility functions are hump-shaped which replicate those that are found in VARs. 47 See Appendix B. 48 It is assumed that there is no population growth in this model.


1. Aggregate per capita consumption has positive drift even though, by definition of REPI, individual consumption is a random walk without drift. 2. The drift in per capita consumption exceeds the drift in per capita labour income whenever the rate of interest is greater than zero (ϕλ > 1 since ϕ > 1 if r > 0). This is a testable implication of the model. 3. Changes in per capita consumption are correlated with lagged innovations in labour income. 4. The drift in per capita consumption and the marginal propensity to consume depend on the age distribution of the economy. For policy makers, this model suggests that it is very important to take into account life cycle considerations.49 The age distribution of an economy is likely to be a crucial variable determining consumption. The change to individual consumption at the individual level after a change in a policy instrument will depend on the age of the consumer. At the aggregate level, it will depend on the age distribution in the economy. For instance, a reduction in interest rates when the majority of the population is near-retirement or retired is unlikely to lead to large increases in aggregate consumption (it may even have the opposite effect). The opposite may be true if the majority of the economy is young. Thus, the sensitivity of aggregate consumption to changes in interest rates is likely to depend on the age distribution in the economy. As we saw in the section examining near-rationality, we find that not only may responses of consumption to changes in instruments be asymmetrical, but that distribution measures (in this case age) may be an important variable in explaining aggregate consumption behaviour in the economy. Another important prediction of the model is that the mpc out of permanent income is less than one. The younger the population, the closer the (aggregate) mpc will be to one. Thus, a government plan to reduce taxes permanently will have a greater impact on consumption and therefore the economy if the population is younger.


This is something that Modigliani had mentioned some 40 years earlier, see section 2.3.


Box D: Risk aversion and prudence In this box risk aversion and prudence are defined. Expressions that determine the strength of risk aversion and precautionary saving are shown. Risk aversion pertains to an individual’s behaviour when facing risk. A consumer is said to be risk averse when he/she always refuses to take fair bets. Denoting a random variable with a ˜ x], we say that an agent above it, and the expected value of such a variable by an overbar, x = E[e is risk averse if x)] U (x) > E [U (e ie agents prefer a payment E[e x] (with certainty that yields utility U (x)) than bearing the risk x e (and obtaining expected utility E [U (e x)]). The strength of risk aversion is governed by rav(x) = −

u00 (x) . u0 (x)

Thus the more concave utility is, the more risk averse an agent is. Some theoretical economists (Pratt, 1964, Arrow, 1965, Kimball, 1990) have argued that it is not unreasonable to think that agents will be less risk averse as their wealth increases, that is, agents face decreasing risk aversion: µ 000 ¶ drav(x) u (x)u0 (x) − u00 (x)2 =− <0 dx u0 (x)2 000


(x) (x) This is guaranteed if − uu00 (x) > − uu0 (x) (requiring u000 (x) > 0). Kimball (1994) defines a precautionary motive as any aspect of an agent’s preferences which causes a risk to affect decisions other than the decision of how strenuously to avoid the risk itself and risks correlated with it (a decision governed by risk aversion). A precautionary motive leads an agent to respond to a risk by making adjustments that help to reduce the expect cost of the risk. Certainty equivalence, which is the absence of precautionary motives, arises when an agent has no way to affect the expected cost of a risk. Kimball (1990) shows that the strength of precautionary saving is determined by prudence:

Ψ(x) = −

u000 (x) u00 (x)

Thus prudence pertains to the third derivative of the utility function. The more convex marginal utility is, the more prudent consumers will be. Decreasing prudence occurs when dΨ(x) dx < 0. What is the difference between risk aversion and prudence? Can both coexist? Loosely speaking risk aversion determines how agents respond to a risk facing them today for which they insure against today. If we believe that decreasing risk aversion characterises agents’ behaviour then prudence will exist. By saving today, agents may use this extra wealth to compensate tomorrow’s risk, thereby paying a lower premium tomorrow. Thus, prudence is any action an agent undertakes today to mitigate the impact of future risk. The agent will still be risk averse today and in the future but can mitigate future risk if he/she can save today. This is why Kimball refers to a precautionary motive as a motive that leads an agent to respond to a risk by making adjustments that help to reduce the expected cost of the risk. Mathematically, the relationship between risk aversion and prudence is given by Ψ(x) − rav(x) = − which is positive if decreasing risk aversion exists. 25

d ln (rav (x)) dx

Diagrammatically we can observe the implications of prudence: Marginal Utility




Precautionary savings






The diagram represents the Euler equation u0 (ct ) = v 0 (ct+1 ) where v0 (ct+1 ) denotes the utility function in ‘present value’ terms. With convex marginal utility, Ev 0 (c) > v 0 (E(c)). In the face of no uncertainty, the optimal level of consumption is c. Introducing uncertainty leads to a lower level of consumption, c∗ . The difference between the level originating with uncertainty, c∗ , and the level without, c, is precautionary saving. The difference depends on the convexity of the marginal utility function which Ψ(ct ) determines, and on the size of uncertainty.



Other consumption issues

In this section we consider the impact that durable goods and aggregation have on consumption modelling and we discuss the implications for policy makers.



In this section we consider the effects of durable consumption. We first examine durables within a rational expectations permanent income hypothesis problem of the type considered in section 4. We then briefly explain a model developed by Caballero (1994) which seeks to explain durable consumption behaviour at both the micro and macro levels. As we shall see, Caballero’s model is not very different from Caballero’s (1995) model of near-rationality that we saw in section 5.2. Introducing durables modifies the consumption problem of section 4 in two ways. First, unlike non-durable goods, durable goods tend to last more than one period and therefore give utility to consumers over a number of periods. Since consumers care about the service flow that a good provides, rather than having the good per se (eg consumers derive satisfaction from driving a car or using a washing machine, not from having the good itself), the argument entering the utility function should relate to the service flow from the durable good and not to its expenditure or stock level. Second, durable goods have a depreciation rate which is less than 100%. Thus, consumers must purchase durable goods when these become fully depreciated. These arguments are considered in a modified version of the consumption problem examined in section 4 (Mankiw, 1982): max V (θKt , θKt+1 , . . . , θKt+T ) = Et

T −t X

(1 + δ)−τ u (θKt+τ )


τ =0

T −t X τ =0


³ ´ (R)−τ cdt+τ − wt+τ = At Kt+1 = (1 − γ) Kt + cdt


where K denotes the stock of durables, θ is a parameter which makes the service flow from durables proportional to the stock of durables (θK is the service flow), γ is the depreciation rate of the durable stock and cd is expenditure on consumption durables.50 The Euler equation for consumption (see Mankiw (1982, p. 419) is 1+δ 0 u (θKt ) (42) Et u0 (θKt+1 ) = 1+r Assuming quadratic preferences the expression for durable expenditure is cdt+1 = cdt + εt+1 − (1 − γ) εt


Consumption expenditure on durable goods obeys an ARIMA(0,1,1), where the moving average component is a function of the rate of depreciation. Obviously, if γ = 1, goods fully depreciate after one period (ie the good is non-durable), and (43) reduces to (16). Intuitively, given that durable goods last for more than a period, an increase in the expenditure of durable goods that increases the stock of durables does not require further increases with the exception of depreciation. There is an interesting implication of this equation: past income surprises, reflected in εt , do affect expenditure on durable goods and thus, with durables we would expect to find that there is excess 50 Mathematically, a difference between this problem and the problem encountered in section 4, is that we now have 2 state variables, K and A, and one control variable cd .


sensitivity. As in section 4, the impact of changes to interest rates on consumption is unknown; it will be negative if the sum of the substitution and wealth effects exceeds the income effect. In the same paper, Mankiw tests whether equation (43) satisfies aggregate US data. He finds that consumption expenditure is better modelled as a random walk suggesting that equation (43) is not able to match US data. Subsequent research by Caballero (1990b) qualified Mankiw’s result that durable consumption expenditure at the aggregate level is fairly smooth. It also documents (Caballero, 1990b, 1994) the rather obvious finding that purchases of durable goods at the microeconomic level are ‘sporadic and lumpy, rather than continuous and smooth.’ [Caballero (1994, p. 108)]. This implies that a representative agent model will not be able to match both micro (lumpy) and macro (smooth) data. Caballero (1994) has suggested that a modification to the basic REPI model for durable consumption expenditure may match both the micro and the aggregate data. This modification comprises a micro-founded model with significant fixed/lumpy adjustment costs. Models with fixed adjustment costs often produce so-called (S,s) rules. The model we examined in section 5.2 was an (S,s) model, and as we shall see Caballero’s model of durable goods with fixed adjustment costs is not very different from the near-rational model we examined earlier. As before, agents define a disequilibrium variable zt ≡ kt − kt∗ , such that if the disequilibrium reaches a lower, L, or upper, U , threshold consumers adjust the stock of durable goods they hold, k. k∗ is the optimal stock of durable goods that would be held if there were no adjustment costs. Various functional forms for k∗ may be considered, such as REPI, or a version of the buffer stock model.51 The dynamics for this economy are very similar to those explained in section 5.2 so will not be pursued here. Nonetheless it is interesting to show how the model of fixed adjustment costs is able to match the aggregate data. It matches the micro data because purchases of durable goods are not continuous but lumpy; agents only purchase durable goods when they reach a trigger point. The arguments that follow are taken from Caballero (1993, pp. 354-5). Take a positive wealth shock. Such shock, increases k∗ but because agents face adjustment costs, leaves k unchanged. The disequilibirum variable, z then becomes negative, implying that the contemporaneous correlation between k ∗ and z is negative. Over time, z increases, generating positive serial correlation in the process for ∆kt . Since changes in the capital stock are the innovations, εt , in (43), it is possible to re-express equation (43) as ∆cdt+1 = (1 − (1 − γ) L) εt+1

= (1 − (1 − γ) L) ∆kt+1 ¤ £ ∗ = (1 − (1 − γ) L) ∆kt+1 + ∆zt+1

∗ and z have a negative contemporaneous correlation, we where L is the lag operator. Since kt+1 will observe that the innovation in aggregate durable consumption expenditure growth will be white noise, as observed in the data. It is worth pointing out that, as before, the cross sectional distribution of agents in the economy is likely to play a crucial role in explaining aggregate durable consumption expenditure. Thus, the implications for policy makers of durable models without precautionary savings or liquidity constraints are similar to those noted in sections 4 and 5.2.


Aggregation issues

As we saw in sections 5.2 on near-rationality, 5.5 on finite lifetimes and 6.1 on durables, the aggregation of individual consumption functions to arrive at an aggregate consumption function 51

Carroll and Dunn (1997) consider a durable consumption model under uncertainty and CCRA preferences such that agents undertake buffer stock behaviour. The ‘adjustment cost’ in that model is the deposit needed to obtain a mortgage to buy a house (the durable good). The model produces (S,s) rules.


can lead to problems if variables such as age distributions or other cross sectional distributions are not considered. In this section we provide further arguments for why the use of a representative agent model may provide bad policy recommendations. We use an example based on Carroll (2000b). When consumption functions are non-linear aggregation is crucial. Consider figure 1 below. It shows the solution to the buffer stock consumption model examined in section 5.1.1 (the concave consumption function called buffer stock) as well as the consumption function that results when preferences are quadratic (the dotted line termed REPI).52 Consumption c, is a concave function of total resources53 (X = A + w). Concavity implies that the marginal propensity to consume decreases as total resources increase, a result of the assumption that preferences exhibit decreasing absolute prudence.54 Why is concavity important? Figure 1 shows that as the level of total resources increases (as we move from point A to point C), the level of consumption increases (we move from point i to point iii) at a decreasing rate (the marginal propensity to consume decreases, M P C A > M P C B > M P C C). The marginal propensity to consume in the case of REPI is constant, suggesting that it is not a function of total resources. Using REPI does not lead to aggregation problems since all agents have the same marginal propensity to consume and therefore, the marginal propensity to consume of a representative agent is the same as that one of an individual agent. Two stylised facts suggest that aggregation is potentially important. Firstly, as we have seen in section 5, REPI does not provide an accurate depiction of consumption behaviour in many countries. Instead, models which incorporate precautionary saving such as the buffer stock model appear to perform better implying that concave consumption functions best model consumption. Secondly, in many countries the wealth distribution is fairly skewed; often the wealthiest 5% of the population own 70% or more of all wealth. Thus the ‘average consumer’ does not hold the ‘average level of wealth’ (the median level of wealth is lower than the mean level of wealth). It is easy to see how a skewed distribution, together with a buffer stock model will imply that a model that considers the average level of wealth and consumption will give wrong policy implications. Assume that a large proportion of consumers (say 90% or more of agents) in the economy are concentrated around point A (‘poor consumers’) in Figure 1 and the remaining fraction (‘rich consumers’) are concentrated around point C. Poor consumers consume i and their marginal propensity to consume is M P C A. Rich consumers consume iii and have a lower propensity to consume M P C C. The average level of wealth in this economy is given (roughly) by point B, yielding a corresponding level of consumption equal to ii and a marginal propensity to consume M P C B. Thus if we were using a representative agent model based on the buffer stock model, we would say that average consumption would be given by ii since the average level of wealth is B. However, the average level of consumption in this economy is not equal to point ii. It is more likely to be given by a point between i and ii, so that the average level of consumption is lower that in the representative economy but the marginal propensity to consume is higher. Thus the average marginal propensity 52 The diagram is a modified version of a figure obtained from using Carroll’s Mathematica programmes that solve the buffer stock model. These programmes are available from http://www.econ.jhu.edu/people/ccarroll/index.html. 53 The arguments in the diagram are actually the ratio of consumption, C, to the permanent component of labour income (the unit root component, equivalent to εt in (24)), c = C/ε, and the ratio of the total resources to the permanent component of labour income, x = X/ε. Dividing the level of consumption and current resources by ε is a convenient mathematical transformation for the dynamic programme as it reduces the number of state variables by 1. As the buffer stock model does not have an exact analytical solution, numerical techniques are used. These techniques can be very computationally costly; reducing the number of state variables simplifies the problem greatly and saves on computer time. See Deaton (1992, pp. 180-94) for an exposition of the use of numerical techniques. Carroll’s website has excellent review material on numerical techniques. Judd (1998) is often cited in the literature. 54 As we saw in section 5.1, the difference between REPI and consumption when there is no certainty equivalence is equal to precautionary saving. Thus, the distance between the concave function and the straight line termed REPI reflects precautionary saving. These decrease as wealth increases, ie there is decreasing absolute prudence.


to consume in this economy is greater than the marginal propensity to consume warranted by the representative agent model. Why is this important? Consider the typical (Keynesian) experiment whereby taxes are reduced to stimulate consumption and therefore aggregate demand. A representative agent model would suggest that a reduction in taxation that leads to an increase in post-tax income of z% will increase consumption by (M P C B) × z%. In reality, the impact of an increase in post-tax labour income will have a greater impact on consumption because the average marginal propensity to consume in this economy is greater than M P CB. Similar predictions can be made about the impact that interest rates changes will have on consumption if the functional form which maps the response of consumption to a change in the interest rate is non-linear.




iii ii i

Buffer stock





Figure 1: Non-linear consumption functions and aggregation The message to draw from these discussions is that, when faced with consumption functions55 that are non-linear, we should be aware that providing policy recommendations based on a representative agent model may give incorrect prescriptions. A direct implication of the above arguments is that it is important to know how wealth is distributed in an economy. Changes in policy instruments are likely to have different effects depending on the impact that they have on those who are rich and those who are less fortunate, and changes in the distribution of wealth are likely to affect aggregate consumption.


Conclusions and suggestions for further reading

Consumption is an important component of most economies. For many countries it constitutes over 50% of GDP. Thus modelling consumption successfully is an important requirement for successful policy making. In this handbook we have considered some of the most important theoretical developments in consumption research over the last 25 years. We have paid attention to the implications that different theories have for policy makers. We have highlighted the importance of forward-looking policy making, uncertainty and the role that interest rates may have on consumption and its growth rate. We have also sketched the implications of using representative agent 55

Or any other function such as an investment function, a Phillips curve, money demand function, etc.


models to explain aggregate consumption behaviour and noted that there can be potential aggregation problems. Consumption research is never stationary; it is one of the most studied areas of macroeconomics. Hence (as the bibliography already notes), there are many papers that can also be read by those interested in consumption research. Deaton (1992) is an excellent starting point.56 Muellbauer (1994), Muellbauer and Lattimore (1995) and Attanasio (1999) have produced remarkable reviews of the literature. Carroll (2001b, 2001c) reviews the main workings of the buffer stock model. Browning and Lusardi (1996) review the literature on saving using many consumption models. Parker (1999) considers different theories of consumption that may explain why the saving ratio was low in the late 1990s in the US. Elmendorf (1996) considers the impact that interest rates have on consumption under a variety of models.

56 Most graduate macroeconomic textbooks have good overviews of the consumption literature and terminology. Romer (1996) is a clear example.



The derivation of the Euler equation

In this appendix we derive the Euler equation.57 For the case of REPI we had the following problem: Consumers wish to maximise V (ct , ct+1 , . . . , ct+T ) = Et

T −t X

(1 + δ)−τ u (ct+τ )


τ =0

subject to

R (At + wt − ct ) = At+1


where R = 1 + r. We can express the problem in terms of the value function Jt (At ) = u (ct ) + (1 + δ)−1 Et Jt+1 (At+1 ) subject to R (At + wt − ct ) = At+1 where Jt (At ) = max V (ct , ct+1 , . . . , ct+T ) {ct }

ie Jt (At ) gives the maximised value of the lifetime utility function V (ct , ct+1 , . . . , ct+T ). If one substitutes the budget constraint (45) into the value function and takes the first order conditions with respect to c and A we get: 0 (R (At + wt − ct )) ct : 0 = u0 (ct ) + Et (1 + δ)−1 Jt+1 0 (R (At + wt − ct )) At : Jt0 (At ) = Et (1 + δ)−1 Jt+1

where we note that

∂At+1 ∂At

∂At+1 ∂ct

∂At+1 ∂At

(46) (47)

= − ∂A∂ct+1 = R. Re-writing the last two expressions as t 0 (At+1 ) R u0 (ct ) = Et (1 + δ)−1 Jt+1


0 (At+1 ) R Jt0 (At ) = Et (1 + δ)−1 Jt+1


we get u0 (ct ) = Jt0 (At ). Leading this expression and substituting into (48) yields u0 (ct ) = Et (1 + δ)−1 Ru0 (ct+1 )


which is the Euler equation reported in equation (12). 57

There are many books explaining dynamic programming. Dixit (1990), Sargent (1987), Lucas and Stokey (1987), Chiang (1994) and Kamien and Schwartz (1991) are good sources. Walsh (1998) and Obstfeld and Rogoff (1996) contain good applied examples.



Innovations in consumption and labour income: tests for excess sensitivity and smoothness

The relationship between labour income and consumption. In the main text we saw that the disturbance term in (16) conveys information about the impact of all new information that becomes available to the consumer in period t about his or her lifetime well-being. All the past/predictable information is reflected in the lagged consumption term. Hall demonstrates that it is possible to derive an expression for that unpredictable element. Non-human wealth evolves according to the expression; At = (1 + r) (At−1 − ct−1 + wt−1 ) and human wealth, Ht , evolves according to Ht = (1 + r) (Ht − wt−1 ) +

T −t X τ =0

(Et wt+τ − Et−1 wt+τ )

so that the behaviour of the total wealth stock is given by the following equation: At + Ht = (1 + r) (At−1 − ct−1 + Ht−1 ) + η t where ηt =

T −t X τ =0

(Et wt+τ − Et−1 wt+τ )

The evolution of total wealth then depends, ceteris paribus, on the relationship between two informational variables, η t and εt . By imposing quadratic utility or certainty equivalence, that relationship is given by: " µ ¶T −t # λ λ + ··· + ηt εt = 1 + 1+r 1+r This is according to Hall ‘the modified annuity value of the increment in wealth. The modification takes account of the consumer’s plans to make consumption grow at a proportional rate λ over the rest of his life.’ (pp. 975-6). The actual form of η t depends on the labour income process. According to REPI, the innovation in consumption is driven by the expectational change in the discounted sum of current and future labour income ∞ X p (1 + r)−i (Et+1 − Et ) wt+i (51) ∆ct+1 = ∆yt+1 = r i=1

This result was first noted by Flavin (1981). If the process for labour income, wt , can be modelled correctly, it should be possible to calculate the innovations in permanent income given by the right hand side of the expression above and compare these to the innovations in consumption (calculated directly from the data).58

B.1 B.1.1

Testing for excess sensitivity and smoothness Excess sensitivity

The test for excess sensitivity (on aggregate time series data) is relatively simple. One only needs to run the following regression: ∆ct = α + β i ∆wt−i + εt , i = 1, 2, ... 58

See Bakhshi (2000) for an interesting example.


and test the restriction that β i = 0, ie that the coefficients for the lagged values of labour income innovations are not significant. If this test is rejected we conclude that there is evidence of excess sensitivity.59 B.1.2

Excess smoothness

Testing for excess smoothness is a little bit more involved. Hansen and Sargent (1981) and Quah (1990) have shown that the innovation in permanent income can be sensitive to the process assumed for labour income. If it is assumed that the income process can be represented by a trend stationary income process wt = B (L) t 60 then we will have r

∞ X i=1

(1 + r)−i (Et+1 − Et ) wt+i = (rρ) B (ρ)


where B(L) denotes the polynomial in the lag operator, and denotes the same polynomial evaluated 1 . On the other hand, if the labour income process is a difference stationary process at L=ρ = 1+r ∆wt = A (L) εt then we will have r

∞ X i=1

(1 + r)−i (Et+1 − Et ) wt+i = A (ρ) εt+1

where A (L) denotes the polynomial Comparing the variance of the innovation ´ ³ in´the ³lag operator. r 1 in consumption with var ( t ) and 1+r B 1+r for a trend stationary income process, or var (εt ) ´ ³ 1 for a difference stationary income process allows a simple test for the existence of and A 1+r excess smoothness. Previous research for US data has reported that difference stationary processes explain the behaviour of labour income better than trend stationary processes. When A and var (εt ) have been estimated on US aggregate time-series data, the implied variance of permanent income has been significantly larger than the variance of consumption changes suggesting that consumption is smoother than predicted by REPI. As Quah points out, ‘this result is remarkably robust across alternative specifications for A’ (fn. 10, p. 457). This result is intuitive; if labour income is nonstationary, innovations to this process will be persistent and will therefore imply a revision to permanent income (and thus consumption) of a similar amount. Since it can be established from the data that the volatility of consumption is less than the volatility of labour income, this being the primary reason Friedman developed the permanent income hypothesis, we find the result that consumption is smoother than permanent income. This fact has been referred to in the literature as the ‘Deaton Paradox’. 59

The test depends on the time series properties of the data, in particular, the labour income term. Flavin (1981) assumed that the labour income process was trend stationary. Deaton (1992), chapter 3, provides an excellent demonstration of spurious regressions using Flavin’s assumption of trend stationarity when the true data generating process is differenced stationary. At the micro/panel data, the tests do not necessarily involve the first difference of the income series, see eg Zeldes (1989), or lagged values of labour income, see eg Attanasio and Weber (1993). 60 ‘We can, without loss, take the trend to be identically zero since here we are interested only in the second-moment properties of consumption and income’ Quah (1990, pp. 455)



The substitution, income and wealth revaluation effects; the two period case with diagrams

In this appendix we show how the impact of a change in interest rates can be decomposed into income, substitution and wealth revaluation effects using a two-period model that can be represented with the use of diagrams. We consider two diagrams, the first one where there is no wealth revaluation, and the second one includes the wealth effect.61 The substitution effect results from the pivot of the budget constraint, which makes consumption cheaper in the future if interest rates increase, thereby enticing consumers to postpone consumption. The income effect results from the change in the present discounted value of consumption (which arises from the fact that consumption is cheaper in the future allowing more consumption today and tomorrow). In both diagrams below, the income effect is labelled "A". The wealth effect results from the change in the present discounted value of income, labelled "B" in the second diagram. Case A: All income is earned in the first period of life, so that the budget constraint is (1 + r)c0 + c1 = (1 + r)x0 . There is no wealth effect, as next period’s income is not ‘revalued’. With an increase in interest rates, the budget constraint pivots (note that if consumers decided not to consume in period 0, they would be able to consume extra in period 1. The dotted line (which is parallel to the new budget constraint) points out the income effect. c1 , x1

x0 , x1

c 0, c 1

c 0, x 0 A

The diagram shows that the original level of income is given at x0 , x1 in the diagram (x1 = 0, since no income is earned in that period). The income effect is given by A. Case B: Income is earned in periods 0 and 1. The budget constraint is given by (1 + r)c0 + c1 = (1 + r)x0 + x1 . 61

These diagrams are copied from Elmendorf’s (1996, p. 82).


c1 , x1

x0 , x1 C

c 0, x 0 B A

Following the increase in interest rates, the budget constraint, shifts and pivots as income is earned over two periods. The dotted line labelled C, demonstrates the previous example, that is, if income were equal to zero in period 1. The income effect is worked out as before (the difference between the new constraint and the dotted line). The difference between the two constraints at the x-axis (which was zero before), constitutes the revaluation of wealth and is the wealth effect.



Taylor-series expansion of the habit model

In this appendix, we show how the Taylor-series expansion to the Euler equation in the habits model was done. We start from the Euler equation (33) £ ¡ £ ¤ ¢¤ £ ¤ uct = Rβ β λuzt+2 − (1 − λ) uct+2 + uct+1 − β λuzt+1 − (1 − λ) uct+1 , and collecting terms we have noting that uzt+i = −γuct+i zct+i t+i ¶¸ · µ ct+1 c c ut = ut+1 Rβ + λγ − (1 − λ) zt+1 µ ¸¶ · ct+2 c −Rβ βut+2 λγ − (1 − λ) zt+2 − (1 − λ) = ξ we can write Defining λγ zct+i t+i £ ¤ ¡ ¢ uct = uct+1 Rβ + ξ t+1 − Rβ βuct+2 ξ t+2

we now take a first-order Taylor-series expansion around a steady state where consumption growth is given by σ and where the consumption to habit stock ratio is given by zc = χ. Thus we have for each of the terms in the Euler equation: ¶ ¶ µ µ σct+i−1 σct+i−1 c c cc +u (ct+i − σct+i−1 ) ut+i (ct+i , zt+i ) = u σct+i−1 , σct+i−1 , χ χ ¶µ ¶ µ σct+i−1 σct+i−1 zt+i − +ucz σct+i−1 , χ χ for i=1, 2. Thus the Euler equation can be written as ¶ ¶ µ · µ ¸ σct σct c c cc ut (ct , ht ) ' (Rβ + ξ) u σct , +u (ct+1 − σct ) σct , χ χ ¶µ ¶ µ σct σct zt+1 − + (Rβ + ξ) ucz σct , χ χ · µ ¸ ¶ ¶ µ σct+1 σct+1 2 c cc +Rβ ξ u σct+1 , +u (ct+2 − σct+1 ) σct+1 , χ χ ¶µ ¶ µ σct+1 σct+1 +Rβ 2 ξucz σct+1 , zt+2 − χ χ which we write as

¶ ¶ µ µ σct σ 2 ct 2 c 2 − Rβ ξu σ ct , (52) + ξ) u σct , χ χ ¶µ ¶ ¶µ ¶ µ µ σct σct σ 2 ct σ 2 ct − (Rβ + ξ) ucz σct , zt+1 − − Rβ 2 ξucz σ 2 ct , zt+2 − χ χ χ χ ¶ ¶ µ µ 2 σct σ ct ' (Rβ + ξ) ucc σct , (ct+1 − σct ) + Rβ 2 ξucc σ 2 ct , (ct+2 − σct+1 ) χ χ uct (ct , ht ) − (Rβ


Note the following ¶ ¶ ¶γ(ρ−1) µ µ µ ct+1 γ(ρ−1) ct+1 −ρ σct = c−ρ = (σc ) uc ct+1 , t t+1 χ χ χ ¶ µ ct = σ γ(ρ−1)−ρ uc ct , χ 37


¶ ¶ ¶ µ µ µ ct+2 ct+1 ct γ(ρ−1)−ρ c 2[γ(ρ−1)−ρ] c =σ =σ u ct+1 , u ct , u ct+2 , χ χ χ c

Note that we also have ¶ ¶γ(ρ−1) ¶ µ µ µ ct+1 γ(ρ−1)−1 −ρ ct+1 −ρ−1 ct+1 cc = −ρct+1 + γ (ρ − 1) ct+1 ct+1 , u χ χ χ ¶γ(ρ−1) ¶ µ µ σct γ(ρ−1)−1 −ρ−1 σct = −ρ (σct ) + γ (ρ − 1) (σct )−ρ χ χ " # µ ¶γ(ρ−1)−1 µ ¶γ(ρ−1) ct −ρ−1 ct −ρ γ(ρ−1)−ρ−1 = σ + γ (ρ − 1) ct −ρct χ χ ¶ µ ct γ(ρ−1)−ρ−1 cc ct , u = σ χ Thus (52) can be written as h i uc (ct , ht ) 1 − σ γ(ρ−1)−ρ (Rβ + ξ) − σ 2[γ(ρ−1)−ρ] Rβ 2 ξ ¶µ ¶ ¶µ ¶ µ µ σct σct σ 2 ct σ 2 ct cz 2 cz 2 zt+1 − − Rβ ξu zt+2 − σct , σ ct , − (Rβ + ξ) u χ χ χ χ ¶h µ i σct ' ucc σct , (ct+1 − σct ) + σ γ(ρ−1)−ρ−1 Rβ 2 ξ (ct+2 − σct+1 ) χ which we write as


h i (ct+1 − σct ) 1 ³ ´ σ γ(ρ−1)−ρ−1 Rβ 2 ξ (ct+2 − σct+1 ) + ct ucc σct , σcχt ct ³ ´³ ´ ³ ´³ 2 zt+1 − σcχt − Rβ 2 ξucz σ 2 ct , σ χct zt+2 − (Rβ + ξ) ucz σct , σcχt ³ ´ ' − ucc σct , σcχt ct £ ¤ uc (ct , ht ) 1 − σ γ(ρ−1)−ρ (Rβ + ξ) − σ 2[γ(ρ−1)−ρ] Rβ 2 ξ ³ ´ + ucc σct , σcχt ct

σ2 ct χ


−σ γ(ρ−1)−ρ Rβ 2 ξ ³ ´ >0 ucc σct , σcχt ¶µ ¶ ¶µ ¶ µ µ σct σct σ 2 ct σ 2 ct cz 2 cz 2 zt = (Rβ + ξ) u zt+1 − − Rβ ξu zt+2 − σct , σ ct , χ χ χ χ Γ=


h i Ω = 1 − σ γ(ρ−1)−ρ (Rβ + ξ) − σ 2[γ(ρ−1)−ρ] Rβ 2 ξ

we have expression (34) in the text.


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[19] Campbell, J. Y. and Mankiw, N. G. (1991),.‘The Response of Consumption to Income: A Cross-Country Investigation’, European Economic Review, 35, pp 723—67. [20] Carroll, C. D. (1992), ‘The Buffer Stock Theory of Saving: Some Macroeconomic Evidence’, Brookings Papers on Economic Activity, 2, pp. 61-156. [21] Carroll, C. D. (1997). ‘Buffer stock saving and the life cycle/permanent income hypothesis’, The Quarterly Journal of Economics, CXII, pp. 1-57. [22] Carroll, C. D. (2000a), ‘Solving consumption models with multiplicative habits,’ Economics Letters, 68, pp. 67-77. [23] Carrol C. D. (2000b), ‘Requiem for the Representative Consumer? Aggegate Implications of Microeconomic Consumption Behaviou’, American Economic Review, Papers and Proceedings, 90, pp 110-15. [24] Carroll, C. D. (2000c) ‘ ‘Risky Habits’ and the Marginal Propensity to Consume Out of Permanent Income’, International Economic Journal, 14, pp. 1-41. [25] Carroll, C. D. (2001a), ‘Precautionary Saving and the Marginal Propensity to Consume Out of Permanent Income’, NBER Working Paper Number W8233. [26] Carroll, C. D. (2001b), ‘A Theory of the Consumption Function, With and Without Liquidity Constraints’, Journal of Economic Perspectives, 15, pp 23-46. [27] Carroll, C. D. (2001c), ‘A Theory of the Consumption Function, With and Without Liquidity Constraints (Expanded Version)’, NBER Working Paper W8387. [28] Carroll, C. D. and Dunn, W. E. (1997), ‘Unemployment Expectations, Jumping (S,s) Triggers, and Household Balance Sheets‘ in NBER Macroeconomics Annual, 1997, Bernanke, B. and Rotemberg, J. (eds.) pp. 165-229. MIT Press. [29] Carroll, C. D. and Kimball, M.S. (1996), ‘On the concavity of the consumption function’, Econometrica, 64, pp. 981-92. [30] Carroll, C. D., Overland, J. R. and Weil, D. N. (2000), ‘Saving and growth with habit formation’, American Economic Review, 90, pp. 341-55. [31] Carroll, C. D. and Weil, D. N. (1994), ‘Saving and Growth: A Reinterpretation’, CarnegieRochester Conference Series on Public Policy, 40, pp. 133-92. [32] Chiang, A. C. (1984), Fundamental methods of mathematical economics, third edition. McGraw-Hill. [33] Chiang, A. C. (1992), Elements of dynamic optimisation. McGraw-Hill. [34] Clarida, R. H. (1991), ‘Aggregate Stochastic Implications of the Life Cycle Hypothesis’, Quaterly Journal of Economics, 106, pp. 851-67. [35] Cochrane, J. H. (1989), ‘The Sensitivity of Tests of the Intertemporal Allocation of Consumption to Near-Rational Alternatives’, American Economic Review, 79, pp. 319-37. [36] Constantinides, G. R. (1990), ‘Habit Formation: A Resolution of the Equity Premium Puzzle’, Journal of Political Economy, 98, pp. 519-543.


[37] Cromb, R. and Fernandez-Corugedo, E. W. (2004), ‘Interest rates and consumption when there is no uncertainty’, mimeo, Bank of England. [38] Davidson, J. E. H., Hendry, D. F., Srba, F.. and Yeo, S. (1978), ‘Econometric Modelling of the Aggregate Time-Series Relationship between Consumers’ Expenditure and Income in the United Kingdom’, Economic Journal, 88, pp. 661-92. [39] Deaton, A. (1987), ‘Life-Cycle Models of Consumption: Is the Evidence Consistent with the Theory?’, in Bewley, T. F. (ed.), Advances in Econometrics, Fifth World Congress, 2. Cambridge University Press, pp. 121-48. [40] Deaton, A. (1991), ‘Saving and liquidity constraints’, Econometrica, 59, pp. 1121-1142. [41] Deaton, A. (1992), Understanding consumption. Clarendon Press. [42] Dixit, A. K. (1990), Optimisation in economic Theory, second edition. Oxford University Press. [43] Elmendorf , D. W. (1996), ‘The effect of interest-rate changes on household saving and consumption: A Survey’, mimeo, Federal Reserve Board. [44] Fernandez-Corugedo, E. W. (2000), Essays on Consumption, Unpublished PhD Thesis, University of Bristol. [45] Fernandez-Corugedo E. W. (2002) ‘Soft liquidity constraints and precautionary saving’, Bank of England Working Paper 158. [46] Fischer, I. (1930), The Theory of Interest. New York: Macmillan. [47] Flavin, M. (1981), ‘The Adjustment of Consumption to Changing Expectations About Future Income’, Journal of Political Economy, 89, pp. 974-1009. [48] Friedman, M. (1957), A Theory of the Consumption Function. Princeton University Press. [49] Furher, J. C. (2000), ‘Habit formation in consumption and its implications for monetary policy models’, American Economic Review, 90, pp. 367-90. [50] Hall, R. E. (1978), ‘Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence’ Journal of Political Economy, 86, pp. 971-87. [51] Hamilton, J. D. (1994), Time Series Analysis, Princeton University Press. [52] Hansen, L. P. and Sargent, T. J. (1981), ‘A Note on Wiener-Kolmogorov Forecasting Formulas for Rational Expectations Models’, Economic Letters, 8, pp. 253-60. [53] Hendry, D. F. and Von ungern-Sternberg, T. (1981), ‘Liquidity and Inflation Effects on Consumers’ Expenditure’. In Essays in The Theory and Measurement Of Consumer Behaviour, Deaton, A. (ed.). Cambridge University Press. [54] Gali, J. (1994), ‘Keeping up with the Joneses: Consumption externalities, portfolio choice and asset prices’, Journal of Money, Credit, and Banking, 26, pp. 1-8. [55] Japelli, T. and Pagano, M. (1989), ‘Consumption and capital market imperfections: An international comparison’, American Economic Review, 79, pp. 1088-1105. [56] Judd, K. (1998), Numerical methods in economics. MIT University Press.


[57] Kamian, M. I. and Schwartz, N. L. (1991), Dynamic Optimisation: The calculus of variations and optimal control in economics and management, second edition. North-Holland. [58] Keynes, J. M. (1936), The General Theory of Employment, Interest and Money. Macmillan Co. [59] Kimball, M. S. (1990a), ‘Precautionary saving in the small and in the large’, Econometrica, 58, pp. 53-73. [60] Kimball, M. S. (1994), ‘Precautionary motives for holding assets’, The New Palgrave Dictionary of Money and Finance. [61] Klein, L. R. and Goldberger, A. S. (1955), An econometric model of the United States 19291952. North-Holland. [62] Kreps, D. M. (1990), A course in microeconomic theory. Harvester-Wheatsheaf. [63] Leland, H. E. (1968), ‘Saving and uncertainty: The precautionary demand’, Quarterly Journal of Economics, 82, pp. 465-73. [64] Lucas, R. E. (1976), ‘Econometric Policy Evaluation: a Critique’. In The Phillips Curve and Labour Markets, K. Brunner and A. H. Meltzer (eds.). North Holland. [65] Lucas, R. E. and Stokey, N. L. (1989), Recursive Methods in economic dynamics. Harvard University Press. [66] Mankiw, N. G. (1982), ‘Hall’s consumption hypothesis and durable goods’, Journal of Monetary Economics, 10, pp. 417-25. [67] Mankiw, N. G. and Reis, R. (2002), ‘Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve’, Quarterly Journal of Economics, vol. 117, pp. 1295-1328. [68] Modigliani, F. (1949), ‘Fluctuations in the Saving-Income Ratio: A Problem in Economic Forecasting’, Studies in Income and Wealth, 11, National Bureau of Economic Research, pp. 371-443. [69] Modigliani, F. (1986), ‘Life cycle, Individual Thrift, and the Wealth of Nations’, American Economic Review, 76, pp. 297-313. [70] Modigliani, F. and Brumberg, R. (1980), ‘Utility and Aggregate Consumption Functions: An Attempt at Integration’ in Abel, A. ed., The Collected Papers of Franco Modigliani. Cambridge: MIT Press. [71] Muellbauer, J. N. (1988), ‘Habits, rationality and myopia in the life cycle consumption func, tion’, Annales d’ economie et de Statistique, 9, pp. 47-70. [72] Muellbauer, J. N. (1994), ‘The Assessment: Consumer Expenditure’, Oxford Review of Economic Policy, 10, pp. 1-41. [73] Muellbauer, J. N. and Lattimore, R. (1994), ‘The Consumption Function: A Theoretical and Empirical Overview’ in Pesaran, H. and Wickens, M. R. (eds) Handbook of Applied Econometrics. [74] Nicholson, W. (1992), Microeconomic theory: Basic principles and extensions, 5th Edition. The Dryden Press. 42

[75] Obstfeld, M. and Rogoff, K. (1996), Foundations of international macroeconomics. MIT Press. [76] Parker, J. (1999) ‘Spendthrift in America? On two decades of decline in the U.S. saving rate’ in Bernanke, B. and Rotemberg, J. (eds.), NBER Macroeconomics Annual, MIT Press, pp 317-70. [77] Pischke, J. S. (1995), ‘Individual Income, Incomplete Information and Aggregate Consumption’, Econometrica, 63, pp. 805-40. [78] Pratt, J. W. (1964), ‘Risk Aversion in the Small and the Large’, Econometrica, 32, pp. 122-36. [79] Quah, D. (1990), ‘Permanent and Transitory Movements in Labor Income: An Explanation for ‘Excess Smoothness’ in Consumption’, Journal of Political Economy, 98, pp. 449-75. [80] Ramsey, F. P. (1928), ‘A Mathematical Theory of Saving’, Economic Journal, 38, pp. 543-59. [81] Romer, D. (1996), Advanced Macroeconomics. McGraw-Hill Advanced Series in Economics. [82] Sadmo, A. (1970), ‘The Effect of Uncertainty on Saving Decisions’, Review of Economic Studies, 37, pp. 353-60. [83] Sargent, T. J. (1987), Dynamic macroeconomic theory. Harvard University Press. [84] Stock, J. H. and Watson, M. W. (2003), ‘Has the business cycle changed? Evidence and Explanations’. Paper presented at the Federal Reserve Bank of Kansas City’s symposium ‘Monetary Policy and Uncertainty’, Jackson Hole, Wyoming. [85] Varian, H. R. (1992), Microeconomic analysis, third edition. Norton and Company. [86] Walsh, C. E. (1998), Monetary Theory and Policy. MIT Press. [87] Zeldes, S. P. (1989), ‘Consumption and Liquidity Constraints: An Empirical Investigation’, Journal of Political Economy, 97, pp. 305-46.


Handbooks The CCBS has continued to add new titles to this series, initiated in 1996. The first 14 are available in Russian, and the first eleven in Spanish. No



1 2 3

Glenn Hoggarth Tony Latter Lionel Price

11 12 13 14

Introduction to monetary policy The choice of exchange rate regime Economic analysis in a central bank: models versus judgement Internal audit in a central bank The management of government debt Primary dealers in government securities markets Basic principles of banking supervision Payment systems Deposit insurance Introduction to monetary operations – revised, 2nd edition Government securities: primary issuance Causes and management of banking crises The retail market for government debt Capital flows

15 16 17

Consolidated supervision Repo of Government Securities Financial Derivatives

18 19 20 21

The Issue of Banknotes* Reserves Management Basic Bond Analysis Banking and Monetary Statistics


Unit Root Testing to Help Model Building


Consumption Theory

4 5 6 7 8 9 10

Christopher Scott Simon Gray Robin McConnachie Derrick Ware David Sheppard Ronald MacDonald Simon Gray and Glenn Hoggarth Simon Gray Tony Latter Robin McConnachie Glenn Hoggarth and Gabriel Sterne Ronald MacDonald Simon Gray Simon Gray and Joanna Place Peter Chartres John Nugee Joanna Place John Thorp and Philip Turnbull Lavan Mahadeva and Paul Robinson Emilio FernandezCorugedo

* Withdrawn from publication. An updated version will be released in due course All CCBS Handbooks can be downloaded from our website www.bankofengland.co.uk/ccbshand.htm

Handbooks: Lecture series As financial markets have become increasingly complex, central bankers' demands for specialised technical assistance and training has risen. This has been reflected in the content of lectures and presentations given by CCBS and Bank staff on technical assistance and training courses. In 1999 we introduced a new series of Handbooks: Lecture Series. The aim of this new series is to make available to a wider audience lectures and presentations of broader appeal. The following are available: No Title



Inflation Targeting: The British Experience

William A Allen


Financial Data needs for Macroprudential Surveillance - E Philip Davis What are the key indicators of risks to domestic Financial Stability? Surplus Liquidity: Implications for Central Banks Joe Ganley Implementing Monetary Policy William A Allen

3 4

Handbooks: Research Series The CCBS begun, in March 2001, to publish Research Papers in Finance. One is available now, and others will follow. No Title



Richhild Moessner

Over the Counter Interest Rate Options

All CCBS Handbooks can be downloaded from our website www.bankofengland.co.uk/ccbshand.htm

BOOKS The CCBS also aims to publish the output from its Research Workshop projects and other research. The following is a list of books published or commissioned by CCBS:Richard Brearley, Alastair Clarke, Charles Goodhart, Juliette Healey, Glenn Hoggarth, David Llewellyn, Chang Shu, Peter Sinclair and Farouk Soussa (2001): Financial Stability and Central Banks – a global perspective, Routledge. Lavan Mahadeva and Gabriel Sterne (eds) (October 2000): Monetary Frameworks in a Global Context, Routledge. (This book includes the report of the 1999 Central Bank Governors symposium and a collection of papers on monetary frameworks issues presented at a CCBS Academic Workshop). Liisa Halme, Christian Hawkesby, Juliette Healey, Indrek Saapar and Farouk Soussa (May 2000): Financial Stability and Central Banks: Selected Issues for Financial Safety Nets and Market Discipline, Centre for Central Banking Studies, Bank of England*. E. Philip Davis, Robert Hamilton, Robert Heath, Fiona Mackie and Aditya Narain (June 1999): Financial Market Data for International Financial Stability, Centre for Central Banking Studies, Bank of England*. Maxwell Fry, Isaack Kilato, Sandra Roger, Krzysztof Senderowicz, David Sheppard, Francisio Solis and John Trundle (1999): Payment Systems in Global Perspective, Routledge. Charles Goodhart, Philipp Hartmann, David Llewellyn, Liliana Rojas-Suárez and Steven Weisbrod (1998): Financial Regulation; Why, how and where now? Routledge. Maxwell Fry, (1997): Emancipating the Banking System and Developing Markets for Government Debt, Routledge. Maxwell Fry, Charles Goodhart and Alvaro Almeida (1996): Central Banking in Developing Countries; Objectives, Activities and Independence, Routledge. Forrest Capie, Charles Goodhart, Stanley Fischer and Norbert Schnadt (1994): The Future of Central Banking; The Tercentenary Symposium of the Bank of England, Cambridge University Press. *These are free publications which are posted on our web site and can be downloaded.

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